44 HFOPDE, chapter 2.4.3

 44.1 problem number 1
 44.2 problem number 2
 44.3 problem number 3
 44.4 problem number 4
 44.5 problem number 5
 44.6 problem number 6
 44.7 problem number 7
 44.8 problem number 8

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44.1 problem number 1

problem number 388

Added January 10, 2019.

Problem 2.4.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tanh (\lambda x) w_y = 0 \]

Mathematica

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

Maple

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

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44.2 problem number 2

problem number 389

Added January 10, 2019.

Problem 2.4.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tanh (\lambda y) w_y = 0 \]

Mathematica

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

Maple

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

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44.3 problem number 3

problem number 390

Added January 10, 2019.

Problem 2.4.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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44.4 problem number 4

problem number 391

Added January 10, 2019.

Problem 2.4.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ \left ( \sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda \right ) \left ( a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}} \right ) }{ \left ( \sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda \right ) \left ( \sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda \right ) }{2}^{-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}} \left ( 4\,i{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) y\lambda \,\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}+4\,i{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){\lambda }^{2}\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ){a}^{2}\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) ya\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,i{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) ya\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda \,\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-6\,i{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda \,\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) ya\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,i{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}-{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) a\lambda \,\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,y{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda +2\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) ya\lambda +2\,y{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}+6\,y{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda -4\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) y{\lambda }^{2}+2\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\lambda -2\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a{\lambda }^{2}+2\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ){a}^{2}\lambda -2\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a-\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) a{\lambda }^{2}+2\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{3}+12\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\lambda +14\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a{\lambda }^{2}-4\,{\mbox{$_2$F$_1$}(-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }},-1/2\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-a+\lambda }{\lambda }};\,-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){\lambda }^{3} \right ) \left ({\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }}} \left ( 4\,i{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){\lambda }^{2}\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) ya\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda \,\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-6\,i{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a\lambda \,\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}+4\,i{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) y\lambda \,\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}-{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) a\lambda \,\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) ya\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}-2\,i{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}+{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ){a}^{2}\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,i{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) ya\sqrt{{a}^{2}+4\,a\lambda -{\lambda }^{2}}-2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}y\cosh \left ( \lambda \,x \right ) a\lambda -2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) ya\lambda -2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\lambda +2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a{\lambda }^{2}-2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ){a}^{2}\lambda +2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a+\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+2\,\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\sinh \left ( \lambda \,x \right ) a{\lambda }^{2}-2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}y\cosh \left ( \lambda \,x \right ){a}^{2}-6\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}y\cosh \left ( \lambda \,x \right ) a\lambda +4\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) y{\lambda }^{2}-2\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{3}-12\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){a}^{2}\lambda -14\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ) a{\lambda }^{2}+4\,{\mbox{$_2$F$_1$}(1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }},1/2\,{\frac{a-\lambda +\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}}{\lambda }};\,{\frac{\sqrt{-{a}^{2}-4\,a\lambda +{\lambda }^{2}}+\lambda }{\lambda }};\,1/2\,{\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }})}\cosh \left ( \lambda \,x \right ){\lambda }^{3} \right ) ^{-1}} \right ) \]

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44.5 problem number 5

problem number 392

Added January 10, 2019.

Problem 2.4.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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44.6 problem number 6

problem number 393

Added January 10, 2019.

Problem 2.4.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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44.7 problem number 7

problem number 394

Added January 10, 2019.

Problem 2.4.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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44.8 problem number 8

problem number 395

Added January 10, 2019.

Problem 2.4.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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