6.4.15 6.1

   6.4.15.1 [1115] Problem 1
   6.4.15.2 [1116] Problem 2
   6.4.15.3 [1117] Problem 3
   6.4.15.4 [1118] Problem 4
   6.4.15.5 [1119] Problem 5

6.4.15.1 [1115] Problem 1

problem number 1115

Added March 9, 2019.

Problem Chapter 4.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.15.2 [1116] Problem 2

problem number 1116

Added March 9, 2019.

Problem Chapter 4.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.15.3 [1117] Problem 3

problem number 1117

Added March 9, 2019.

Problem Chapter 4.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.15.4 [1118] Problem 4

problem number 1118

Added March 9, 2019.

Problem Chapter 4.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.15.5 [1119] Problem 5

problem number 1119

Added March 9, 2019.

Problem Chapter 4.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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