86 HFOPDE, chapter 3.6.5

 86.1 Problem 1
 86.2 Problem 2
 86.3 Problem 3
 86.4 Problem 4
 86.5 Problem 5
 86.6 Problem 6

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86.1 Problem 1

problem number 749

Added Feb. 11, 2019.

Problem Chapter 3.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{kx}{a}}-{\frac{1}{\mu \,b\lambda \,a} \left ( -{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \mu \,b\lambda \,a-c\sin \left ({\frac{\mu \, \left ( ay-bx \right ) }{a}}+{\frac{\mu \,bx}{a}} \right ) a\lambda +\cos \left ( \lambda \,x \right ) \mu \,b \right ) } \]

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86.2 Problem 2

problem number 750

Added Feb. 11, 2019.

Problem Chapter 3.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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86.3 Problem 3

problem number 751

Added Feb. 11, 2019.

Problem Chapter 3.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{2 a^3 \lambda ^2 c_1\left (\frac{a y-b x}{a}\right )-a^2 \lambda \cos \left (\mu \left (\frac{a y-b x}{a}+\frac{b x}{a}\right )+\lambda x\right )+2 a^2 c \lambda ^2 x-a^2 \lambda \cos (\lambda x-\mu y)-2 a b^2 \mu ^2 c_1\left (\frac{a y-b x}{a}\right )-a b \mu \cos (\lambda x-\mu y)+a b \mu \cos \left (\mu \left (\frac{a y-b x}{a}+\frac{b x}{a}\right )+\lambda x\right )-2 b^2 c \mu ^2 x}{2 a (a \lambda -b \mu ) (a \lambda +b \mu )}\right \}\right \} \]

Maple

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

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86.4 Problem 4

problem number 752

Added Feb. 11, 2019.

Problem Chapter 3.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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86.5 Problem 5

problem number 753

Added Feb. 11, 2019.

Problem Chapter 3.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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86.6 Problem 6

problem number 754

Added Feb. 11, 2019.

Problem Chapter 3.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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