42 HFOPDE, chapter 2.4.1

 42.1 problem number 1
 42.2 problem number 2
 42.3 problem number 3
 42.4 problem number 4
 42.5 problem number 5
 42.6 problem number 6
 42.7 problem number 7

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42.1 problem number 1

problem number 373

Added January 10, 2019.

Problem 2.4.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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42.2 problem number 2

problem number 374

Added January 10, 2019.

Problem 2.4.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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42.3 problem number 3

problem number 375

Added January 10, 2019.

Problem 2.4.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{e^{-\frac{a e^{-\lambda x}}{\lambda }} \left (-2 y e^{\frac{a e^{-\lambda x}}{\lambda }+\lambda x} \int _1^{e^{\lambda x}} \frac{e^{\frac{a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]} \, dK[1]+a e^{\frac{a e^{-\lambda x}}{\lambda }} \int _1^{e^{\lambda x}} \frac{e^{\frac{a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]} \, dK[1]+a e^{\frac{a e^{-\lambda x}}{\lambda }+2 \lambda x} \int _1^{e^{\lambda x}} \frac{e^{\frac{a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]} \, dK[1]-2 \lambda e^{\frac{a e^{\lambda x}}{\lambda }+\lambda x}\right )}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}\right )\right \}\right \} \]

Maple

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

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42.4 problem number 4

problem number 376

Added January 10, 2019.

Problem 2.4.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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42.5 problem number 5

problem number 377

Added January 10, 2019.

Problem 2.4.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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42.6 problem number 6

problem number 378

Added January 10, 2019.

Problem 2.4.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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42.7 problem number 7

problem number 379

Added January 10, 2019.

Problem 2.4.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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