52 HFOPDE, chapter 2.6.4

 52.1 problem number 1
 52.2 problem number 2
 52.3 problem number 3
 52.4 problem number 4
 52.5 problem number 5
 52.6 problem number 6
 52.7 problem number 7
 52.8 problem number 8
 52.9 problem number 9
 52.10 problem number 10
 52.11 problem number 11
 52.12 problem number 12

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52.1 problem number 1

problem number 473

Added January 20, 2019.

Problem 2.6.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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52.2 problem number 2

problem number 474

Added January 20, 2019.

Problem 2.6.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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52.3 problem number 3

problem number 475

Added January 20, 2019.

Problem 2.6.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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52.4 problem number 4

problem number 476

Added January 20, 2019.

Problem 2.6.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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52.5 problem number 5

problem number 477

Added January 20, 2019.

Problem 2.6.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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52.6 problem number 6

problem number 478

Added January 20, 2019.

Problem 2.6.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{\sqrt{b^2} y-a \sqrt{b^2} \cot (a x)}{b^2}\right )-\sqrt{b^2} x\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{{{\rm e}^{-2\,ibx}} \left ( ia\cot \left ( ax \right ) -iy-b \right ) }{b \left ( ib-a\cot \left ( ax \right ) +y \right ) }} \right ) \]

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52.7 problem number 7

problem number 479

Added January 20, 2019.

Problem 2.6.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\lambda x) w_x + a \cot (\mu y) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 \cos (\mu y) \cos ^{-\frac{a \mu }{\lambda }}(\lambda x)}{\mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac{1}{\lambda \,\mu } \left ( \ln \left ({\frac{ \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1}{ \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}}} \right ) a\mu +\lambda \,\ln \left ( \left ( \cos \left ( \mu \,y \right ) \right ) ^{2} \right ) \right ) } \right ) \]

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52.8 problem number 8

problem number 480

Added January 20, 2019.

Problem 2.6.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\mu y) w_x + a \cot (\lambda x) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 \sin (\mu y) \sin ^{-\frac{a \mu }{\lambda }}(\lambda x)}{\mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{a\mu }\ln \left ({\frac{\tan \left ( \mu \,y \right ) }{ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1}\sqrt{ \left ( \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \left ( -2\, \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{-1} \right ) ^{{\frac{a\mu }{\lambda }}}}} \right ) } \right ) \]

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52.9 problem number 9

problem number 481

Added January 20, 2019.

Problem 2.6.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\mu y) w_x + a \cot ^2(\lambda x) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{4 \sin (\mu y) e^{\frac{a \mu (\lambda x+\cot (\lambda x))}{\lambda }}}{\mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{a\mu }\ln \left ({{\rm e}^{-{\frac{a\mu \,\pi }{\lambda }}}}\sqrt{{\frac{1}{ \left ( \cos \left ( \mu \,y \right ) \right ) ^{2}}{{\rm e}^{2\,a\mu \,x+{\frac{a\mu \,\pi }{\lambda }}+2\,{\frac{\cos \left ( \lambda \,x \right ) a\mu }{\lambda \,\sin \left ( \lambda \,x \right ) }}}}}}\sin \left ( \mu \,y \right ) \cos \left ( \mu \,y \right ) \right ) } \right ) \]

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52.10 problem number 10

problem number 482

Added January 20, 2019.

Problem 2.6.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (y+a) w_x + c \cot (x+b) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (4 \sin (a+y) e^{c (\cot (b+x)+x)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{1}{\tan \left ( b \right ) \left ( \tan \left ( x \right ) +\tan \left ( b \right ) \right ) } \left ( -\pi \,\tan \left ( b \right ) \tan \left ( x \right ) c-\pi \, \left ( \tan \left ( b \right ) \right ) ^{2}c+2\,\tan \left ( b \right ) \tan \left ( x \right ) cx-2\, \left ( \tan \left ( b \right ) \right ) ^{2}\tan \left ( x \right ) c+2\, \left ( \tan \left ( b \right ) \right ) ^{2}cx-\ln \left ( \left ( \sin \left ( y \right ) \right ) ^{-2} \right ) \tan \left ( b \right ) \tan \left ( x \right ) -\ln \left ( \left ( \sin \left ( y \right ) \right ) ^{-2} \right ) \left ( \tan \left ( b \right ) \right ) ^{2}+2\,\ln \left ({\frac{\cos \left ( y \right ) \tan \left ( a \right ) +\sin \left ( y \right ) }{\sin \left ( y \right ) \tan \left ( a \right ) }} \right ) \tan \left ( b \right ) \tan \left ( x \right ) +2\,\ln \left ({\frac{\cos \left ( y \right ) \tan \left ( a \right ) +\sin \left ( y \right ) }{\sin \left ( y \right ) \tan \left ( a \right ) }} \right ) \left ( \tan \left ( b \right ) \right ) ^{2}-2\,c\tan \left ( x \right ) \right ) } \right ) \]

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52.11 problem number 11

problem number 483

Added January 20, 2019.

Problem 2.6.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\lambda x) \cot (\mu y) w_x + a w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 \sin (\mu y) \cos ^{\frac{a \mu }{\lambda }}(\lambda x)}{\mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{a\mu }\ln \left ({\frac{\tan \left ( \mu \,y \right ) }{ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1}\sqrt{ \left ( \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2\,{\frac{a\mu }{\lambda }}}}} \right ) } \right ) \]

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52.12 problem number 12

problem number 484

Added January 20, 2019.

Problem 2.6.4.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\lambda x) \cot (\mu y) w_x + a \cot (v x) w_y = 0 \]

Mathematica

\[ \text{Timed out} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{a\mu }\ln \left ({{\it csgn} \left ( \left ( \cos \left ( \mu \,y \right ) \right ) ^{-1} \right ) \sin \left ( \mu \,y \right ){{\rm e}^{a\mu \,x}} \left ({{\rm e}^{2\,ivx}}-1 \right ) ^{{\frac{ia\mu }{v}}} \left ({{\rm e}^{a\mu \,\int \!-2\,{\frac{{{\rm e}^{2\,ivx}}+1}{ \left ({{\rm e}^{2\,ivx}}-1 \right ) \left ({{\rm e}^{2\,i\lambda \,x}}+1 \right ) }}\,{\rm d}x}} \right ) ^{-1}} \right ) } \right ) \]