51 HFOPDE, chapter 2.6.3

 51.1 problem number 1
 51.2 problem number 2
 51.3 problem number 3
 51.4 problem number 4
 51.5 problem number 5
 51.6 problem number 6
 51.7 problem number 7
 51.8 problem number 8
 51.9 problem number 9
 51.10 problem number 10
 51.11 problem number 11
 51.12 problem number 12
 51.13 problem number 13
 51.14 problem number 14
 51.15 problem number 15

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51.1 problem number 1

problem number 458

Added January 14, 2019.

Problem 2.6.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-a \lambda x-b \lambda x+\log (\cos (\lambda x))+\lambda y}{\lambda }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac{-2\,xa\lambda -2\,bx\lambda +2\,\lambda \,y-\ln \left ( 1+ \left ( \tan \left ( \lambda \,x \right ) \right ) ^{2} \right ) }{\lambda }} \right ) \]

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51.2 problem number 2

problem number 459

Added January 14, 2019.

Problem 2.6.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-x+\frac{2 (a+b) \tan ^{-1}(\tan (\lambda y))+2 \log (a+b+\tan (\lambda y))-\log \left (\sec ^2(\lambda y)\right )}{2 \lambda (a+b-i) (a+b+i)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac{2\,{a}^{2}\lambda \,x+4\,ab\lambda \,x+2\,{b}^{2}\lambda \,x-2\,\arctan \left ( \tan \left ( \lambda \,y \right ) \right ) a-2\,\arctan \left ( \tan \left ( \lambda \,y \right ) \right ) b+2\,\lambda \,x+\ln \left ( 1+ \left ( \tan \left ( \lambda \,y \right ) \right ) ^{2} \right ) -2\,\ln \left ( a+\tan \left ( \lambda \,y \right ) +b \right ) }{\lambda \, \left ({a}^{2}+2\,ab+{b}^{2}+1 \right ) }} \right ) \]

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51.3 problem number 3

problem number 460

Added January 14, 2019.

Problem 2.6.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.4 problem number 4

problem number 461

Added January 14, 2019.

Problem 2.6.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.5 problem number 5

problem number 462

Added January 14, 2019.

Problem 2.6.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.6 problem number 6

problem number 463

Added January 14, 2019.

Problem 2.6.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a x y \tan ^k(b x)+a \tan ^k(b x)+y^2\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \] Timed out

Maple

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

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51.7 problem number 7

problem number 464

Added January 14, 2019.

Problem 2.6.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.8 problem number 8

problem number 465

Added January 20, 2019.

Problem 2.6.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +\left ( a \tan ^n(\lambda x) y^2- a b^2 \tan ^{n+2}(\lambda x) + b \lambda \tan ^2(\lambda x)+ b \lambda \right ) w_y = 0 \]

Mathematica

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

Maple

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

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51.9 problem number 9

problem number 466

Added January 20, 2019.

Problem 2.6.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.10 problem number 10

problem number 467

Added January 20, 2019.

Problem 2.6.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.11 problem number 11

problem number 468

Added January 20, 2019.

Problem 2.6.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{k+y} \left ( \left ( k+y \right ) \int \!{\frac{\cos \left ( \lambda \,x \right ) }{b\cos \left ( \lambda \,x \right ) +a\sin \left ( \lambda \,x \right ) } \left ({{\rm e}^{2\,i\mu \,x}}+1 \right ) ^{{\frac{-ic}{ \left ( a+ib \right ) \mu }}} \left ( \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-2} \right ) ^{{\frac{ak}{\lambda \, \left ({a}^{2}+{b}^{2} \right ) }}} \left ({\frac{b\cos \left ( \lambda \,x \right ) +a\sin \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) ^{-2\,{\frac{ak}{\lambda \, \left ({a}^{2}+{b}^{2} \right ) }}}{{\rm e}^{{\frac{1}{ \left ({a}^{2}+{b}^{2} \right ) \left ( a+ib \right ) \lambda } \left ( 2\, \left ( a+ib \right ) \lambda \,ac \left ({a}^{2}+{b}^{2} \right ) \int \!{\frac{{{\rm e}^{2\,i\mu \,x}}-1}{ \left ( \left ( a+ib \right ){{\rm e}^{2\,i\lambda \,x}}+ib-a \right ) \left ( a+ib \right ) \left ({{\rm e}^{2\,i\mu \,x}}+1 \right ) }}\,{\rm d}x-2\,k \left ( a+ib \right ) b\arctan \left ({\frac{\sin \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) -cx\lambda \, \left ({a}^{2}+{b}^{2} \right ) \right ) }}}}\,{\rm d}x+ \left ({{\rm e}^{2\,i\mu \,x}}+1 \right ) ^{{\frac{-ic}{ \left ( a+ib \right ) \mu }}} \left ( \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-2} \right ) ^{{\frac{ak}{\lambda \, \left ({a}^{2}+{b}^{2} \right ) }}} \left ({\frac{b\cos \left ( \lambda \,x \right ) +a\sin \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) ^{-2\,{\frac{ak}{\lambda \, \left ({a}^{2}+{b}^{2} \right ) }}}{{\rm e}^{{\frac{1}{ \left ({a}^{2}+{b}^{2} \right ) \left ( a+ib \right ) \lambda } \left ( 2\, \left ( a+ib \right ) \lambda \,ac \left ({a}^{2}+{b}^{2} \right ) \int \!{\frac{{{\rm e}^{2\,i\mu \,x}}-1}{ \left ( \left ( a+ib \right ){{\rm e}^{2\,i\lambda \,x}}+ib-a \right ) \left ( a+ib \right ) \left ({{\rm e}^{2\,i\mu \,x}}+1 \right ) }}\,{\rm d}x-2\,k \left ( a+ib \right ) b\arctan \left ({\frac{\sin \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) -cx\lambda \, \left ({a}^{2}+{b}^{2} \right ) \right ) }}} \right ) } \right ) \]

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51.12 problem number 12

problem number 469

Added January 20, 2019.

Problem 2.6.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^m x^n+b x\right )+w^{(0,1)}(x,y) \tan ^k(\lambda y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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51.13 problem number 13

problem number 470

Added January 20, 2019.

Problem 2.6.3.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.14 problem number 14

problem number 471

Added January 20, 2019.

Problem 2.6.3.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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51.15 problem number 15

problem number 472

Added January 20, 2019.

Problem 2.6.3.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a x^n \tan ^m(y)+b x\right )+w^{(0,1)}(x,y) \tan ^k(\lambda y)=0,w(x,y),\{x,y\}\right ] \] Timed out

Maple

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