113 HFOPDE, chapter 4.6.5

 113.1 Problem 1
 113.2 Problem 2
 113.3 Problem 3
 113.4 Problem 4
 113.5 Problem 5
 113.6 Problem 6

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113.1 Problem 1

problem number 927

Added March 9, 2019.

Problem Chapter 4.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = (b \sin (\lambda x)+k \cos (\mu y)) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1(y-a x) e^{\frac{k \sin (\mu y)}{a \mu }-\frac{b \cos (\lambda x)}{\lambda }}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ){{\rm e}^{{\frac{-b\cos \left ( \lambda \,x \right ) \mu \,a+k\sin \left ( \mu \,y \right ) \lambda }{\lambda \,\mu \,a}}}} \]

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113.2 Problem 2

problem number 928

Added March 9, 2019.

Problem Chapter 4.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \sin (\mu y) w_y = (b \sin (\lambda x)+k \tan (\mu y)) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1(y-a x) e^{-\frac{b \cos (\lambda x)}{\lambda }} \cos ^{-\frac{k}{a \mu }}(\mu y)\right \}\right \} \]

Maple

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

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113.3 Problem 3

problem number 929

Added March 9, 2019.

Problem Chapter 4.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \sin (\mu y) w_y = b \tan (\lambda x) w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\mu \,a}\ln \left ( \RootOf \left ( \mu \,y-\arctan \left ( 2\,{\frac{{\it \_Z}\,{{\rm e}^{a\mu \,x}}}{{{\it \_Z}}^{2}{{\rm e}^{2\,a\mu \,x}}+1}},-{\frac{{{\it \_Z}}^{2}{{\rm e}^{2\,a\mu \,x}}-1}{{{\it \_Z}}^{2}{{\rm e}^{2\,a\mu \,x}}+1}} \right ) \right ) \right ) } \right ) \left ( 1+ \left ( \tan \left ( \lambda \,x \right ) \right ) ^{2} \right ) ^{1/2\,{\frac{b}{\lambda }}} \]

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113.4 Problem 4

problem number 930

Added March 9, 2019.

Problem Chapter 4.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tan (\mu y) w_y = b \sin (\lambda x) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to e^{-\frac{b \cos (\lambda x)}{\lambda }} c_1\left (\frac{\log (\sin (\mu y))-a \mu x}{\mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{\ln \left ({{\rm e}^{-a\mu \,x}}{\it csgn} \left ( \left ( \cos \left ( \mu \,y \right ) \right ) ^{-1} \right ) \sin \left ( \mu \,y \right ) \right ) }{\mu \,a}} \right ){{\rm e}^{-{\frac{b\cos \left ( \lambda \,x \right ) }{\lambda }}}} \]

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113.5 Problem 5

problem number 931

Added March 9, 2019.

Problem Chapter 4.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \sin (\lambda x) w_x + a w_y = b \cos (\mu y) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to e^{\frac{b \sin (\mu y)}{a \mu }} c_1\left (\frac{-a \log \left (\sin \left (\frac{\lambda x}{2}\right )\right )+a \log \left (\cos \left (\frac{\lambda x}{2}\right )\right )+\lambda y}{\lambda }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{\lambda \,y-a\ln \left ( \csc \left ( \lambda \,x \right ) -\cot \left ( \lambda \,x \right ) \right ) }{\lambda }} \right ){{\rm e}^{{\frac{b\sin \left ( \mu \,y \right ) }{\mu \,a}}}} \]

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113.6 Problem 6

problem number 932

Added March 9, 2019.

Problem Chapter 4.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\lambda x) w_x + a w_y = b \tan (\mu y) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to \cos ^{-\frac{b}{a \mu }}(\mu y) c_1\left (\frac{a \log (\cos (\lambda x))}{\lambda }+y\right )\right \}\right \} \]

Maple

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