49 HFOPDE, chapter 2.6.1

 49.1 problem number 1
 49.2 problem number 2
 49.3 problem number 3
 49.4 problem number 4
 49.5 problem number 5
 49.6 problem number 6
 49.7 problem number 7
 49.8 problem number 8
 49.9 problem number 9
 49.10 problem number 10
 49.11 problem number 11
 49.12 problem number 12
 49.13 problem number 13
 49.14 problem number 14

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49.1 problem number 1

problem number 432

Added January 14, 2019.

Problem 2.6.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a \sin ^k(\lambda x) + b \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{\sin ^2(\lambda x)^{-\frac{k}{2}-\frac{1}{2}} \left (-a \cos (\lambda x) \sin ^{k+1}(\lambda x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-k}{2},\frac{3}{2},\cos ^2(\lambda x)\right )+b \lambda x \sin ^2(\lambda x)^{\frac{k}{2}+\frac{1}{2}}-\lambda y \sin ^2(\lambda x)^{\frac{k}{2}+\frac{1}{2}}\right )}{\lambda }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx+y-\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x \right ) \] contains unresolved integral

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49.2 problem number 2

problem number 433

Added January 14, 2019.

Problem 2.6.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a \sin ^k(\lambda y) + b \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \frac{1}{a \sin ^k(\lambda K[1])+b} \, dK[1]-x\right )\right \}\right \} \] contains unresolved integral

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \sin \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \] contains unresolved integral

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49.3 problem number 3

problem number 434

Added January 14, 2019.

Problem 2.6.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a \sin ^k(\lambda y) \sin ^n(\mu y) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1()\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac{ \left ( \sin \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] contains unresolved integral

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49.4 problem number 4

problem number 435

Added January 14, 2019.

Problem 2.6.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a \sin ^k(x+\lambda y) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac{y\lambda +x}{\lambda }}}\! \left ( 1+a \left ( \sin \left ({\it \_a}\,\lambda \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}\lambda +x \right ) \] contains unresolved integral

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49.5 problem number 5

problem number 436

Added January 14, 2019.

Problem 2.6.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +\left (y^2-a^2 + a \lambda \sin (\lambda x)+a^2 \sin ^2(\lambda x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a^2 \sin ^2(\lambda x)-a^2+a \lambda \sin (\lambda x)+y^2\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\sqrt{2\,{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) +2} \left ( 2\, \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunC \left ( 4\,{\frac{a}{\lambda }},-1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) a+ \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunCPrime \left ( 4\,{\frac{a}{\lambda }},-1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) \lambda +2\,y\HeunC \left ( 4\,{\frac{a}{\lambda }},-1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \right ) \left ( 2\,\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunC \left ( 4\,{\frac{a}{\lambda }},1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) a+\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunCPrime \left ( 4\,{\frac{a}{\lambda }},1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) \lambda +2\, \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunC \left ( 4\,{\frac{a}{\lambda }},1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) a+ \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunCPrime \left ( 4\,{\frac{a}{\lambda }},1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) \lambda +2\,y\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunC \left ( 4\,{\frac{a}{\lambda }},1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) +{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunC \left ( 4\,{\frac{a}{\lambda }},1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) \lambda +2\,y\HeunC \left ( 4\,{\frac{a}{\lambda }},1/2,-1/2,-2\,{\frac{a}{\lambda }},1/8\,{\frac{8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \right ) ^{-1}} \right ) \]

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49.6 problem number 6

problem number 437

Added January 14, 2019.

Problem 2.6.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +\left ( y^2 + a \sin (\beta x) y + a b \sin (\beta x)-b^2 \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a b \sin (\beta x)+a y \sin (\beta x)-b^2+y^2\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{b+y} \left ( b\int \!{{\rm e}^{-{\frac{2\,bx\beta +a\cos \left ( \beta \,x \right ) }{\beta }}}}\,{\rm d}x+y\int \!{{\rm e}^{-{\frac{2\,bx\beta +a\cos \left ( \beta \,x \right ) }{\beta }}}}\,{\rm d}x+{{\rm e}^{-{\frac{2\,bx\beta +a\cos \left ( \beta \,x \right ) }{\beta }}}} \right ) } \right ) \] contains unresolved integrals

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49.7 problem number 7

problem number 438

Added January 14, 2019.

Problem 2.6.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +\left ( y^2 + a x \sin ^m(b x) y + a \sin ^m(b x)\right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a x y \sin ^m(b x)+a \sin ^m(b x)+y^2\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac{a \left ( \sin \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{a \left ( \sin \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac{a \left ( \sin \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \] contains unresolved integrals

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49.8 problem number 8

problem number 439

Added January 14, 2019.

Problem 2.6.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +\left (\lambda \sin (\lambda x) y^2 + \lambda \sin ^3(\lambda x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (\lambda y^2 \sin (\lambda x)+\lambda \sin ^3(\lambda x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{\sqrt{\pi } \left ( y+\cos \left ( \lambda \,x \right ) \right ) }{\sqrt{\pi }\erfi \left ( \cos \left ( \lambda \,x \right ) \right ) y+\sqrt{\pi }\erfi \left ( \cos \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) -2\,{{\rm e}^{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}}}}} \right ) \]

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49.9 problem number 9

problem number 440

Added January 14, 2019.

Problem 2.6.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ 2 w_x +\left ((\lambda +a-a \sin (\lambda x)) y^2 + \lambda -a -a \sin (\lambda x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (y^2 (a (-\sin (\lambda x))+a+\lambda )-a \sin (\lambda x)-a+\lambda \right )+2 w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1} \left ( -y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{3}-y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{3}+3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{a}^{2}\lambda -3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{3}-3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}-y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\lambda }^{3}-5\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda -2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}+y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{3}+y\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+y\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\lambda }^{3}+7\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +5\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}-3\,y\sin \left ( \lambda \,x \right ){a}^{2}\lambda -3\,y\sin \left ( \lambda \,x \right ) a{\lambda }^{2}+ \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\lambda }^{3}- \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda -3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}-3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}+3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\lambda }^{3}-y\sin \left ( \lambda \,x \right ){a}^{3}-y\sin \left ( \lambda \,x \right ){\lambda }^{3}+y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{a}^{3}+y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}-3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}-y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +3\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}+5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda +3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}-2\,\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda -4\,\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}-3\,\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda -2\,\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}-5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}\lambda -3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) a{\lambda }^{2}+2\,\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda +4\,\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}+3\,\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}\lambda +2\,\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) a{\lambda }^{2}-2\,\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\lambda }^{3}+ \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda +2\, \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}+2\,\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{3}- \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda -2\, \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{3}-\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}+\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}+5\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}-y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}-4\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{2}\lambda -y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-7\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda -5\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}+2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{2}\lambda +y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+3\,y\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +3\,y\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}-2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +4\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}-2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda -y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2} \right ) \left ( -2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{4}+2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{4}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}-6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda - \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\lambda }^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\lambda }^{4}-2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\lambda }^{4}-6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}\lambda -14\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}{\lambda }^{2}-10\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{3}-10\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}{\lambda }^{2}-4\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{3}+2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}\lambda -2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}\lambda -2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}{\lambda }^{2}-6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda -8\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}{\lambda }^{2}-2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{3}+6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda +14\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}{\lambda }^{2}+10\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){\lambda }^{3}-2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda +2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda +2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\lambda }^{3}+6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda +4\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{2}\lambda +4\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda -3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}-5\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}-3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) a{\lambda }^{2}+3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda +5\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) a{\lambda }^{2}-2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}\lambda -4\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}{\lambda }^{2}-2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) a{\lambda }^{3}+2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}\lambda +6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}{\lambda }^{2}+6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{3}-6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}\lambda -4\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}{\lambda }^{2}+6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{3}\lambda +10\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}{\lambda }^{2}+4\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{3}+2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda +4\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}{\lambda }^{2}+2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{3}-2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}\lambda -6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}{\lambda }^{2}-6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{3}+6\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{3}\lambda +8\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ){a}^{2}{\lambda }^{2}+2\,{{\rm e}^{{\frac{a\sin \left ( \lambda \,x \right ) }{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) a{\lambda }^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ){a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) a{\lambda }^{2}+3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}a{\lambda }^{2}-3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ({\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ){a}^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\lambda }^{3}-4\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a{\lambda }^{2}+3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){a}^{2}\lambda +3\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ) a{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}\sin \left ( \lambda \,x \right ){\lambda }^{3}+ \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}{a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt{\sin \left ( \lambda \,x \right ) +1}y\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac{{\it \_a}\,a-a-\lambda }{ \left ({\it \_a}-1 \right ) ^{3/2}\sqrt{{\it \_a}+1}}{{\rm e}^{{\frac{{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{a}^{3} \right ) ^{-1}} \right ) \]

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49.10 problem number 10

problem number 441

Added January 14, 2019.

Problem 2.6.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +\left ((\lambda +a \sin ^2(\lambda x)) y^2 + \lambda -a +a \sin ^2(\lambda x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (y^2 \left (a \sin ^2(\lambda x)+\lambda \right )+a \sin ^2(\lambda x)-a+\lambda \right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1} \left ( 2\,y \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a-4\,y\cos \left ( 2\,\lambda \,x \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a+\sin \left ( 2\,\lambda \,x \right ) \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}a+2\,y \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}\lambda +2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a-2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) a-2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) \lambda -4\,y\cos \left ( 2\,\lambda \,x \right ) \lambda +a\sin \left ( 2\,\lambda \,x \right ) +2\,\sin \left ( 2\,\lambda \,x \right ) \lambda +2\,y\lambda \right ) \left ( 8\,\sin \left ( 2\,\lambda \,x \right ){{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt{-1+\cos \left ( 2\,\lambda \,x \right ) }{\lambda }^{2}+\sin \left ( 2\,\lambda \,x \right ) \sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa+2\,\sin \left ( 2\,\lambda \,x \right ) \sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda +2\,y\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda +2\,y \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-4\,y\cos \left ( 2\,\lambda \,x \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-4\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ){{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt{-1+\cos \left ( 2\,\lambda \,x \right ) }a\lambda +\sin \left ( 2\,\lambda \,x \right ) \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa+2\,y \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda +2\,y \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) \sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) \sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda +4\,\sin \left ( 2\,\lambda \,x \right ){{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt{-1+\cos \left ( 2\,\lambda \,x \right ) }a\lambda -4\,y\cos \left ( 2\,\lambda \,x \right ) \sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac{ \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \lambda \,\sin \left ( 2\,\lambda \,x \right ) }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt{\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac{\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda \right ) ^{-1}} \right ) \]

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49.11 problem number 11

problem number 442

Added January 14, 2019.

Problem 2.6.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x -\left ( (k+1) x^k y^2 - a x^{k+1}(\sin x)^m y + a (\sin x)^m \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ \text{sol = ()} \] Timed out

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49.12 problem number 12

problem number 443

Added January 14, 2019.

Problem 2.6.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +\left ( a \sin ^k(\lambda x + \mu )(y-b x^n -c)^2 + y - b x^n + b n x^{n-1} - c \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ \text{sol = ()} \] Timed out

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49.13 problem number 13

problem number 444

Added January 14, 2019.

Problem 2.6.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x +\left ( a \sin ^m(\lambda x ) y^2 + k y + a b^2 x^{2 k} \sin ^m(\lambda x) \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{y x^{-k}}{\sqrt{b^2}}\right )-\sqrt{b^2} \int _1^x a K[1]^{k-1} \sin ^m(\lambda K[1]) \, dK[1]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( ab\int \!{x}^{k-1} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{m}\,{\rm d}x-\arctan \left ({\frac{y{x}^{-k}}{b}} \right ) \right ) \]

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49.14 problem number 14

problem number 445

Added January 14, 2019.

Problem 2.6.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a \sin (\lambda x) + b) w_x +\left ( y^2+ c \sin (\mu x) y - k^2 + c k \sin (\mu x) \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{k+y} \left ( k\int \!{\frac{1}{a\sin \left ( \lambda \,x \right ) +b}{{\rm e}^{{\frac{1}{\lambda \,\sqrt{-{a}^{2}+{b}^{2}}} \left ( c\int \!{\frac{\sin \left ( \mu \,x \right ) }{a\sin \left ( \lambda \,x \right ) +b}}\,{\rm d}x\lambda \,\sqrt{-{a}^{2}+{b}^{2}}-4\,k\arctan \left ({\frac{b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\sqrt{-{a}^{2}+{b}^{2}}\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) \right ) }}}}\,{\rm d}x+y\int \!{\frac{1}{a\sin \left ( \lambda \,x \right ) +b}{{\rm e}^{{\frac{1}{\lambda \,\sqrt{-{a}^{2}+{b}^{2}}} \left ( c\int \!{\frac{\sin \left ( \mu \,x \right ) }{a\sin \left ( \lambda \,x \right ) +b}}\,{\rm d}x\lambda \,\sqrt{-{a}^{2}+{b}^{2}}-4\,k\arctan \left ({\frac{b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\sqrt{-{a}^{2}+{b}^{2}}\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) \right ) }}}}\,{\rm d}x+{{\rm e}^{{\frac{1}{\lambda \,\sqrt{-{a}^{2}+{b}^{2}}} \left ( c\int \!{\frac{\sin \left ( \mu \,x \right ) }{a\sin \left ( \lambda \,x \right ) +b}}\,{\rm d}x\lambda \,\sqrt{-{a}^{2}+{b}^{2}}-4\,k\arctan \left ({\frac{b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\sqrt{-{a}^{2}+{b}^{2}}\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) \right ) }}} \right ) } \right ) \]