6.4.16 6.2

   6.4.16.1 [1120] Problem 1
   6.4.16.2 [1121] Problem 2
   6.4.16.3 [1122] Problem 3
   6.4.16.4 [1123] Problem 4
   6.4.16.5 [1124] Problem 5

6.4.16.1 [1120] Problem 1

problem number 1120

Added March 9, 2019.

Problem Chapter 4.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.16.2 [1121] Problem 2

problem number 1121

Added March 9, 2019.

Problem Chapter 4.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.16.3 [1122] Problem 3

problem number 1122

Added March 9, 2019.

Problem Chapter 4.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.16.4 [1123] Problem 4

problem number 1123

Added March 9, 2019.

Problem Chapter 4.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cos ^n(\lambda x) w_y = (c \cos ^m(\mu x)+s \cos ^k(\beta y)) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{b \sqrt{\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(\lambda x)\right )}{a \lambda n+a \lambda }+y\right ) \exp \left (\int _1^x\frac{s \cos ^k\left (\frac{\beta \left (b \csc (\lambda x) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(\lambda x)\right ) \sqrt{\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+a \lambda (n+1) y-b \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(\lambda K[1])\right ) \sqrt{\sin ^2(\lambda K[1])}\right )}{a \lambda (n+1)}\right )+c \cos ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac{b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( c \left ( \cos \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( \cos \left ( \beta \, \left ( -\int \!{\frac{b \left ( \cos \left ({\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}+\int \!{\frac{b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-y \right ) \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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6.4.16.5 [1124] Problem 5

problem number 1124

Added March 9, 2019.

Problem Chapter 4.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cos ^n(\lambda y) w_y = (c \cos ^m(\mu x)+s \cos ^k(\beta y)) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{\sqrt{\sin ^2(\lambda y)} \csc (\lambda y) \cos ^{1-n}(\lambda y) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(\lambda y)\right )}{\lambda (n-1)}-\frac{b x}{a}\right ) \exp \left (\int _1^y\frac{\cos ^{-n}(\lambda K[1]) \left (s \cos ^k(\beta K[1])+c \cos ^m\left (\frac{\mu \left (a \csc (\lambda y) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(\lambda y)\right ) \sqrt{\sin ^2(\lambda y)} \cos ^{1-n}(\lambda y)-b \lambda (n-1) x-a \cos ^{1-n}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(\lambda K[1])\right ) \sqrt{\sin ^2(\lambda K[1])}\right )}{b \lambda (n-1)}\right )\right )}{b}dK[1]\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-a\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y+bx}{b}} \right ){{\rm e}^{\int ^{y}\!{\frac{ \left ( \cos \left ({\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \cos \left ({\frac{\mu }{b} \left ( a\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y-\int \!{\frac{ \left ( \cos \left ({\it \_b}\,\lambda \right ) \right ) ^{-n}a}{b}}\,{\rm d}{\it \_b}b-bx \right ) } \right ) \right ) ^{m}+s \left ( \cos \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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