46 HFOPDE, chapter 2.4.5

 46.1 problem number 1
 46.2 problem number 2
 46.3 problem number 3
 46.4 problem number 4
 46.5 problem number 5
 46.6 problem number 6

_______________________________________________________________________________________

46.1 problem number 1

problem number 400

Added January 10, 2019.

Problem 2.4.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{2 \left (a \mu \cosh (\lambda x)-2 \lambda \tan ^{-1}\left (\tanh \left (\frac{\mu y}{2}\right )\right )\right )}{\lambda \mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{\cosh \left ( \lambda \,x \right ) a\mu -2\,\arctan \left ({{\rm e}^{\mu \,y}} \right ) \lambda }{\lambda \,a\mu }} \right ) \]

_______________________________________________________________________________________

46.2 problem number 2

problem number 401

Added January 10, 2019.

Problem 2.4.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{{{\rm e}^{\lambda \,x}}a\mu -a\mu \,{{\rm e}^{-\lambda \,x}}+4\,\arctanh \left ({{\rm e}^{\mu \,y}} \right ) \lambda }{\lambda \,a\mu }} \right ) \]

_______________________________________________________________________________________

46.3 problem number 3

problem number 402

Added January 10, 2019.

Problem 2.4.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-4 \lambda x} \left (16 \lambda ^2 x e^{4 \lambda x}+16 \lambda ^2 x e^{8 \lambda x}+8 \lambda x y e^{4 \lambda x}-8 \lambda x y e^{8 \lambda x}-y e^{4 \lambda x}-y e^{8 \lambda x}+y e^{12 \lambda x}+14 \lambda e^{4 \lambda x}-14 \lambda e^{8 \lambda x}+2 \lambda e^{12 \lambda x}-2 \lambda +y\right )}{2 \left (-y e^{4 \lambda x}+2 \lambda e^{4 \lambda x}+2 \lambda +y\right )}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 4\,{( \left ({\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\sinh \left ( \lambda \,x \right ) \lambda +{\rm coth} \left (\lambda \,x\right )y\sinh \left ( \lambda \,x \right ) -2\,{\rm coth} \left (\lambda \,x\right )\cosh \left ( \lambda \,x \right ) \lambda -\sinh \left ( \lambda \,x \right ) \lambda ) \left ( -4\, \left ({\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\sinh \left ( \lambda \,x \right ) \ln \left ({\frac{\cosh \left ( \lambda \,x \right ) -\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda +4\, \left ({\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\sinh \left ( \lambda \,x \right ) \ln \left ({\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda + \left ({\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\cosh \left ( 3\,\lambda \,x \right ) \lambda - \left ({\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\cosh \left ( 5\,\lambda \,x \right ) \lambda -4\,{\rm coth} \left (\lambda \,x\right )y\sinh \left ( \lambda \,x \right ) \ln \left ({\frac{\cosh \left ( \lambda \,x \right ) -\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) +4\,{\rm coth} \left (\lambda \,x\right )y\sinh \left ( \lambda \,x \right ) \ln \left ({\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) +8\,{\rm coth} \left (\lambda \,x\right )\cosh \left ( \lambda \,x \right ) \ln \left ({\frac{\cosh \left ( \lambda \,x \right ) -\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda -8\,{\rm coth} \left (\lambda \,x\right )\cosh \left ( \lambda \,x \right ) \ln \left ({\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda +6\,\sinh \left ( 3\,\lambda \,x \right ){\rm coth} \left (\lambda \,x\right )\lambda -2\,\sinh \left ( 5\,\lambda \,x \right ){\rm coth} \left (\lambda \,x\right )\lambda +{\rm coth} \left (\lambda \,x\right )y\cosh \left ( 3\,\lambda \,x \right ) -{\rm coth} \left (\lambda \,x\right )y\cosh \left ( 5\,\lambda \,x \right ) +8\,{\rm coth} \left (\lambda \,x\right )\sinh \left ( \lambda \,x \right ) \lambda +4\,\sinh \left ( \lambda \,x \right ) \ln \left ({\frac{\cosh \left ( \lambda \,x \right ) -\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda -4\,\sinh \left ( \lambda \,x \right ) \ln \left ({\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda -\cosh \left ( 3\,\lambda \,x \right ) \lambda +\cosh \left ( 5\,\lambda \,x \right ) \lambda \right ) ^{-1}} \right ) \]

_______________________________________________________________________________________

46.4 problem number 4

problem number 403

Added January 10, 2019.

Problem 2.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({ \left ( -2\,a-3\,\lambda \right ) \left ( y\cosh \left ( \lambda \,x \right ) \sinh \left ( \lambda \,x \right ) -a \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}- \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}b+a \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2} \left ({\frac{\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) ^{-{\frac{2\,a+\lambda }{\lambda }}} \left ( - \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{-2} \right ) ^{{\frac{a+b}{\lambda }}} \left ( 2\,y \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{3}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}\sinh \left ( \lambda \,x \right ) a+3\,y \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{3}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}\sinh \left ( \lambda \,x \right ) \lambda +2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{a}^{2}+2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}ab+3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\lambda +3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}b\lambda -2\,\cosh \left ( \lambda \,x \right ) y{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\sinh \left ( \lambda \,x \right ) -3\,\cosh \left ( \lambda \,x \right ) y{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}\lambda \,\sinh \left ( \lambda \,x \right ) -4\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{a}^{2}-2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}ab-8\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\lambda -3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}b\lambda -3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{\lambda }^{2}+4\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(2,-1/2\,{\frac{2\,b-3\,\lambda }{\lambda }};\,1/2\,{\frac{2\,a+5\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}b\lambda -2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(2,-1/2\,{\frac{2\,b-3\,\lambda }{\lambda }};\,1/2\,{\frac{2\,a+5\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{\lambda }^{2}+2\,{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{a}^{2}+5\,{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\lambda +3\,{\mbox{$_2$F$_1$}(1,-1/2\,{\frac{2\,b-\lambda }{\lambda }};\,1/2\,{\frac{2\,a+3\,\lambda }{\lambda }};\,{\frac{ \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{\lambda }^{2} \right ) ^{-1}} \right ) \]

_______________________________________________________________________________________

46.5 problem number 5

problem number 404

Added January 10, 2019.

Problem 2.4.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \sinh (\lambda y) w_x + a \cosh (\beta x) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-\sinh \left ( \beta \,x \right ) a\lambda +\cosh \left ( y\lambda \right ) \beta }{a\beta \,\lambda }} \right ) \]

_______________________________________________________________________________________

46.6 problem number 6

problem number 405

Added January 10, 2019.

Problem 2.4.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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