6.4.19 6.5

   6.4.19.1 [1135] Problem 1
   6.4.19.2 [1136] Problem 2
   6.4.19.3 [1137] Problem 3
   6.4.19.4 [1138] Problem 4
   6.4.19.5 [1139] Problem 5
   6.4.19.6 [1140] Problem 6

6.4.19.1 [1135] Problem 1

problem number 1135

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

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

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6.4.19.2 [1136] Problem 2

problem number 1136

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

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

Maple

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

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6.4.19.3 [1137] Problem 3

problem number 1137

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

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

Maple

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

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6.4.19.4 [1138] Problem 4

problem number 1138

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

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

Maple

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

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6.4.19.5 [1139] Problem 5

problem number 1139

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

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

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6.4.19.6 [1140] Problem 6

problem number 1140

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

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

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