45 HFOPDE, chapter 2.4.4

 45.1 problem number 1
 45.2 problem number 2
 45.3 problem number 3
 45.4 problem number 4

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45.1 problem number 1

problem number 396

Added January 10, 2019.

Problem 2.4.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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45.2 problem number 2

problem number 397

Added January 10, 2019.

Problem 2.4.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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45.3 problem number 3

problem number 398

Added January 10, 2019.

Problem 2.4.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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45.4 problem number 4

problem number 399

Added January 10, 2019.

Problem 2.4.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ({\rm coth} \left (\lambda \,x\right )\LegendreP \left ({\frac{a}{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) a+{\rm coth} \left (\lambda \,x\right )\LegendreP \left ({\frac{a}{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \lambda +y\LegendreP \left ({\frac{a}{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) +\LegendreP \left ({\frac{a+\lambda }{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \sqrt{{a}^{2}+{\lambda }^{2}}-\LegendreP \left ({\frac{a+\lambda }{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) a-\LegendreP \left ({\frac{a+\lambda }{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \lambda \right ) \left ({\rm coth} \left (\lambda \,x\right )\LegendreQ \left ({\frac{a}{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) a+{\rm coth} \left (\lambda \,x\right )\LegendreQ \left ({\frac{a}{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \lambda +y\LegendreQ \left ({\frac{a}{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) +\LegendreQ \left ({\frac{a+\lambda }{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \sqrt{{a}^{2}+{\lambda }^{2}}-\LegendreQ \left ({\frac{a+\lambda }{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) a-\LegendreQ \left ({\frac{a+\lambda }{\lambda }},{\frac{\sqrt{{a}^{2}+{\lambda }^{2}}}{\lambda }},{\rm coth} \left (\lambda \,x\right ) \right ) \lambda \right ) ^{-1}} \right ) \]