50 HFOPDE, chapter 2.6.2

 50.1 problem number 1
 50.2 problem number 2
 50.3 problem number 3
 50.4 problem number 4
 50.5 problem number 5
 50.6 problem number 6
 50.7 problem number 7
 50.8 problem number 8
 50.9 problem number 9
 50.10 problem number 10
 50.11 problem number 11
 50.12 problem number 12

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50.1 problem number 1

problem number 446

Added January 14, 2019.

Problem 2.6.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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50.2 problem number 2

problem number 447

Added January 14, 2019.

Problem 2.6.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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50.3 problem number 3

problem number 448

Added January 14, 2019.

Problem 2.6.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \cos ^{-k}(\lambda K[1]) \cos ^{-n}(\mu K[1]) \, dK[1]-a x\right )\right \}\right \} \]

Maple

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

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50.4 problem number 4

problem number 449

Added January 14, 2019.

Problem 2.6.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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50.5 problem number 5

problem number 450

Added January 14, 2019.

Problem 2.6.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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50.6 problem number 6

problem number 451

Added January 14, 2019.

Problem 2.6.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{{{\sl M}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-\sin \left ( \lambda \,x \right ) y{{\sl M}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )}-1}{2+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{{\sl U}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )}-\sin \left ( \lambda \,x \right ) y{{\sl U}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )}}} \right ) \]

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50.7 problem number 7

problem number 452

Added January 14, 2019.

Problem 2.6.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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50.8 problem number 8

problem number 453

Added January 14, 2019.

Problem 2.6.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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50.9 problem number 9

problem number 454

Added January 14, 2019.

Problem 2.6.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n y^m+b x) w_x +\cos ^k(\lambda y) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \right ) \]

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50.10 problem number 10

problem number 455

Added January 14, 2019.

Problem 2.6.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n +b x \cos ^m y) w_x +y^k w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({x}^{-n+1}{{\rm e}^{b\int \!{y}^{-k} \left ( \cos \left ( y \right ) \right ) ^{m}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \!{y}^{-k} \left ( \cos \left ( y \right ) \right ) ^{m}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \!{y}^{-k} \left ( \cos \left ( y \right ) \right ) ^{m}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y \right ) \]

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50.11 problem number 11

problem number 456

Added January 14, 2019.

Problem 2.6.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n +b x \cos ^m y) w_x + \cos ^k(\lambda y) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \right ) \]

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50.12 problem number 12

problem number 457

Added January 14, 2019.

Problem 2.6.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n \cos ^m y + b x) w_x + \cos ^k(\lambda y) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \right ) \]