43 HFOPDE, chapter 2.4.2

 43.1 problem number 1
 43.2 problem number 2
 43.3 problem number 3
 43.4 problem number 4
 43.5 problem number 5
 43.6 problem number 6
 43.7 problem number 7
 43.8 problem number 8

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43.1 problem number 1

problem number 380

Added January 10, 2019.

Problem 2.4.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \left ( \cosh (\lambda x)\right )w_y = 0 \]

Mathematica

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

Maple

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

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43.2 problem number 2

problem number 381

Added January 10, 2019.

Problem 2.4.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \left ( \cosh (\lambda x)\right )w_y = 0 \]

Mathematica

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

Maple

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

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43.3 problem number 3

problem number 382

Added January 10, 2019.

Problem 2.4.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({ \left ( 8\,y \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{6}a-8\,y \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}\lambda -\sinh \left ( 2\,\lambda \,x \right ) \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}a-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 -a\sinh \left ( 2\,\lambda \,x \right ) +2\,\lambda \,\sinh \left ( 2\,\lambda \,x \right ) \right ) \sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) } \left ( -8\,\int \!2\,{\frac{ \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xy \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{6}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }a+8\,\int \!2\,{\frac{ \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xy \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }\lambda +\int \!2\,{\frac{ \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sinh \left ( 2\,\lambda \,x \right ) \left ( \cosh \left ( 2\,\lambda \,x \right ) \right ) ^{2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }a-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{\cosh \left ( 2\,\lambda \,x \right ) +1}a\lambda +2\,\int \!2\,{\frac{ \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sinh \left ( 2\,\lambda \,x \right ) \cosh \left ( 2\,\lambda \,x \right ) \sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }a-2\,\int \!2\,{\frac{ \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sinh \left ( 2\,\lambda \,x \right ) \cosh \left ( 2\,\lambda \,x \right ) \sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }\lambda -4\,\sinh \left ( 2\,\lambda \,x \right ){{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt{\cosh \left ( 2\,\lambda \,x \right ) +1}a\lambda +8\,\sinh \left ( 2\,\lambda \,x \right ){{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt{\cosh \left ( 2\,\lambda \,x \right ) +1}{\lambda }^{2}+\int \!2\,{\frac{ \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sinh \left ( 2\,\lambda \,x \right ) \sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }a-2\,\int \!2\,{\frac{ \left ( \cosh \left ( 2\,\lambda \,x \right ) a+a-2\,\lambda \right ) \lambda \,\sinh \left ( 2\,\lambda \,x \right ) }{ \left ( \cosh \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{1/2\,{\frac{\cosh \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sinh \left ( 2\,\lambda \,x \right ) \sqrt{-1+\cosh \left ( 2\,\lambda \,x \right ) }\lambda \right ) ^{-1}} \right ) \]

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43.4 problem number 4

problem number 383

Added January 10, 2019.

Problem 2.4.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\sqrt{\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2} \left ( y\cosh \left ( \lambda \,x \right ) +y-\sinh \left ( \lambda \,x \right ) \right ) \left ( \sqrt{\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{5/2}\int \!{\frac{ \left ( a-\lambda +a\cosh \left ( \lambda \,x \right ) \right ) \lambda \,\sinh \left ( \lambda \,x \right ) }{\sqrt{\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{{\frac{a\cosh \left ( \lambda \,x \right ) }{\lambda }}}}}\,{\rm d}xy-\sqrt{\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}\int \!{\frac{ \left ( a-\lambda +a\cosh \left ( \lambda \,x \right ) \right ) \lambda \,\sinh \left ( \lambda \,x \right ) }{\sqrt{\cosh \left ( \lambda \,x \right ) -1} \left ( \cosh \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{{\frac{a\cosh \left ( \lambda \,x \right ) }{\lambda }}}}}\,{\rm d}x\sinh \left ( \lambda \,x \right ) +2\,{{\rm e}^{{\frac{a\cosh \left ( \lambda \,x \right ) }{\lambda }}}}\cosh \left ( \lambda \,x \right ) \sinh \left ( \lambda \,x \right ) \lambda +2\,{{\rm e}^{{\frac{a\cosh \left ( \lambda \,x \right ) }{\lambda }}}}\sinh \left ( \lambda \,x \right ) \lambda \right ) ^{-1}} \right ) \]

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43.5 problem number 5

problem number 384

Added January 10, 2019.

Problem 2.4.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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43.6 problem number 6

problem number 385

Added January 10, 2019.

Problem 2.4.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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43.7 problem number 7

problem number 386

Added January 10, 2019.

Problem 2.4.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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43.8 problem number 8

problem number 387

Added January 10, 2019.

Problem 2.4.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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