6.4.6 3.1

   6.4.6.1 [1060] Problem 1
   6.4.6.2 [1061] Problem 2
   6.4.6.3 [1062] Problem 3
   6.4.6.4 [1063] Problem 4
   6.4.6.5 [1064] Problem 5
   6.4.6.6 [1065] Problem 6
   6.4.6.7 [1066] Problem 7
   6.4.6.8 [1067] Problem 8
   6.4.6.9 [1068] Problem 9
   6.4.6.10 [1069] Problem 10

6.4.6.1 [1060] Problem 1

problem number 1060

Added Feb. 23, 2019.

Problem Chapter 4.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c e^{\alpha x+ \beta y} w \]

Mathematica

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

Maple

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

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6.4.6.2 [1061] Problem 2

problem number 1061

Added Feb. 23, 2019.

Problem Chapter 4.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c e^{\lambda x}+ k e^{\mu y}) w \]

Mathematica

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

Maple

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

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6.4.6.3 [1062] Problem 3

problem number 1062

Added Feb. 23, 2019.

Problem Chapter 4.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b e^{\beta y} w_y = c w \]

Mathematica

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

Maple

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

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6.4.6.4 [1063] Problem 4

problem number 1063

Added Feb. 23, 2019.

Problem Chapter 4.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda y} w_x + b e^{\beta x} w_y = c w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{e^{\lambda y}}{\lambda }-\frac{b e^{\beta x}}{a \beta }\right ) \exp \left (\frac{c \left (\beta x-\log \left (\frac{a \beta e^{\lambda y}}{\lambda }\right )\right )}{a \beta e^{\lambda y}-b \lambda e^{\beta x}}\right )\right \}\right \}\]

Maple

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

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6.4.6.5 [1064] Problem 5

problem number 1064

Added Feb. 23, 2019.

Problem Chapter 4.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b e^{\beta x} w_y = c e^{\gamma y} w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{b e^{x (\beta -\lambda )}}{a (\lambda -\beta )}+y\right ) \exp \left (\int _1^x\frac{c \exp \left (y \gamma -\frac{b \left (e^{(\beta -\lambda ) x}-e^{(\beta -\lambda ) K[1]}\right ) \gamma }{a (\beta -\lambda )}-\lambda K[1]\right )}{a}dK[1]\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-b{{\rm e}^{x \left ( \beta -\lambda \right ) }}+ay \left ( \beta -\lambda \right ) }{ \left ( \beta -\lambda \right ) a}} \right ){{\rm e}^{\int ^{x}\!{\frac{c}{a}{{\rm e}^{1/8\,{\frac{-b{{\rm e}^{x \left ( \beta -\lambda \right ) }}+{{\rm e}^{{\it \_a}\, \left ( \beta -\lambda \right ) }}b+a \left ( \beta -\lambda \right ) \left ( -8\,{\it \_a}\,\lambda +y \right ) }{ \left ( \beta -\lambda \right ) a}}}}}{d{\it \_a}}}}\]

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6.4.6.6 [1065] Problem 6

problem number 1065

Added Feb. 23, 2019.

Problem Chapter 4.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b e^{\beta y} w_y = (c e^{\gamma y} + s e^{\delta y} ) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{b e^{-\lambda x}}{a \lambda }-\frac{e^{-\beta y}}{\beta }\right ) \exp \left (-\frac{e^{-\lambda x} \left (e^{-\beta y}\right )^{-\frac{\delta +\gamma }{\beta }} \left ((\beta -\delta ) \left (c \gamma \left (e^{-\beta y}\right )^{\frac{\delta }{\beta }} \left (\frac{a \lambda e^{\lambda x-\beta y}}{b \beta }\right )^{\frac{\gamma }{\beta }} \, _2F_1\left (\frac{\beta +\gamma }{\beta },\frac{\gamma }{\beta }-1;\frac{\gamma }{\beta };1-\frac{a e^{\lambda x-\beta y} \lambda }{b \beta }\right )+(\beta -\gamma ) \left (c \left (e^{-\beta y}\right )^{\frac{\delta }{\beta }}+s \left (e^{-\beta y}\right )^{\frac{\gamma }{\beta }}\right )\right )+\delta s (\beta -\gamma ) \left (e^{-\beta y}\right )^{\frac{\gamma }{\beta }} \left (\frac{a \lambda e^{\lambda x-\beta y}}{b \beta }\right )^{\frac{\delta }{\beta }} \, _2F_1\left (\frac{\beta +\delta }{\beta },\frac{\delta }{\beta }-1;\frac{\delta }{\beta };1-\frac{a e^{\lambda x-\beta y} \lambda }{b \beta }\right )\right )}{a \lambda (\beta -\delta ) (\beta -\gamma )}\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ \left ({{\rm e}^{\beta \,y}}b\beta -a\lambda \,{{\rm e}^{\lambda \,x}} \right ){{\rm e}^{-\beta \,y-\lambda \,x}}}{b\beta \,\lambda }} \right ){{\rm e}^{-8\,{\frac{ \left ( a\lambda \,{{\rm e}^{\lambda \,x}}-{{\rm e}^{\beta \,y}}b\beta \right ){{\rm e}^{-\beta \,y-\lambda \,x}}+{{\rm e}^{-\lambda \,x}}\beta \,b}{b \left ( 8\,\beta -1 \right ) a\lambda \, \left ( \beta -\delta \right ) } \left ( c \left ( \beta -\delta \right ) \left ({\frac{a\lambda }{ \left ( a\lambda \,{{\rm e}^{\lambda \,x}}-{{\rm e}^{\beta \,y}}b\beta \right ){{\rm e}^{-\beta \,y-\lambda \,x}}+{{\rm e}^{-\lambda \,x}}\beta \,b}} \right ) ^{1/8\,{\beta }^{-1}}+ \left ({\frac{a\lambda }{ \left ( a\lambda \,{{\rm e}^{\lambda \,x}}-{{\rm e}^{\beta \,y}}b\beta \right ){{\rm e}^{-\beta \,y-\lambda \,x}}+{{\rm e}^{-\lambda \,x}}\beta \,b}} \right ) ^{{\frac{\delta }{\beta }}}s \left ( -1/8+\beta \right ) \right ) }}}\]

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6.4.6.7 [1066] Problem 7

problem number 1066

Added Feb. 23, 2019.

Problem Chapter 4.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\beta x} w_x + (b e^{\gamma x}+c e^{\lambda y} ) w_y = (s e^{\mu x} + k e^{\delta y} + p ) w \]

Mathematica

Failed

Maple

\[w \left ( x,y \right ) ={{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( \left ({a \left ( a\int \!{\frac{c}{a}{{\rm e}^{{\frac{-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x\lambda -\int \!{{\rm e}^{{\frac{-8\,b\lambda \,{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}-8\,\beta \,{\it \_b}\, \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}\,{\rm d}{\it \_b}c\lambda +a{{\rm e}^{-8\,{\frac{ \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) ^{-1}} \right ) ^{{\frac{\delta }{\lambda }}}k{{\rm e}^{{\frac{-8\,\delta \,b{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}-8\,\beta \,{\it \_b}\, \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}+s{{\rm e}^{{\it \_b}\, \left ( -\beta +\mu \right ) }}+{{\rm e}^{-\beta \,{\it \_b}}}p \right ) }{d{\it \_b}}}}{\it \_F1} \left ({\frac{1}{\lambda } \left ( -\lambda \,\int \!{\frac{c}{a}{{\rm e}^{{\frac{-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x-{{\rm e}^{-8\,{\frac{ \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) } \right ) \]

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6.4.6.8 [1067] Problem 8

problem number 1067

Added Feb. 23, 2019.

Problem Chapter 4.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\beta x} w_x + (b e^{\gamma x}+c e^{\lambda y} ) w_y = (s e^{\mu x+\delta y} + k ) w \]

Mathematica

Failed

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\lambda } \left ( -\lambda \,\int \!{\frac{c}{a}{{\rm e}^{{\frac{-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x-{{\rm e}^{-8\,{\frac{ \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) } \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( \left ({a \left ( a\int \!{\frac{c}{a}{{\rm e}^{{\frac{-8\,\lambda \,b{{\rm e}^{-\beta \,x+x/8}}-8\,\beta \,x \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}}\,{\rm d}x\lambda -\int \!{{\rm e}^{{\frac{-8\,b\lambda \,{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}-8\,\beta \,{\it \_b}\, \left ( -1/8+\beta \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}\,{\rm d}{\it \_b}c\lambda +a{{\rm e}^{-8\,{\frac{ \left ( b{{\rm e}^{-\beta \,x+x/8}}+ay \left ( -1/8+\beta \right ) \right ) \lambda }{ \left ( 8\,\beta -1 \right ) a}}}} \right ) ^{-1}} \right ) ^{{\frac{\delta }{\lambda }}}s{{\rm e}^{{\frac{-8\,\delta \,b{{\rm e}^{-\beta \,{\it \_b}+{\it \_b}/8}}+8\,{\it \_b}\, \left ( -1/8+\beta \right ) \left ( -\beta +\mu \right ) a}{ \left ( 8\,\beta -1 \right ) a}}}}+{{\rm e}^{-\beta \,{\it \_b}}}k \right ) }{d{\it \_b}}}}\]

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6.4.6.9 [1068] Problem 9

problem number 1068

Added Feb. 23, 2019.

Problem Chapter 4.3.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\beta x} w_x + b e^{\gamma x+\lambda y} w_y = (c e^{\mu x+\delta y} + k ) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{b e^{\gamma x-\beta x}}{a \beta -a \gamma }-\frac{e^{-\lambda y}}{\lambda }\right ) \exp \left (\frac{c (\gamma -\beta ) \left (e^{\lambda y}\right )^{\delta /\lambda } e^{-\gamma x-\lambda y+\mu x} \, _2F_1\left (1,\frac{\mu -\gamma }{\beta -\gamma };\frac{\beta \delta -\gamma \delta -\gamma \lambda +\lambda \mu }{\beta \lambda -\gamma \lambda };1-\frac{a e^{\beta x-\gamma x-\lambda y} (\beta -\gamma )}{b \lambda }\right )}{b (\beta (\lambda -\delta )+\delta \gamma -\lambda \mu )}-\frac{k e^{-\beta x}}{a \beta }\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ( -8\,{\frac{ \left ( \left ( -1/8+\beta \right ) a{{\rm e}^{\beta \,x-x/8}}-{{\rm e}^{y\lambda }}\lambda \,b \right ){{\rm e}^{-\beta \,x-y\lambda +x/8}}}{b\lambda \, \left ( 8\,\beta -1 \right ) }} \right ){{\rm e}^{\int ^{x}\!{\frac{{{\rm e}^{-\beta \,{\it \_a}}}}{a} \left ( c \left ({\frac{a \left ( 8\,\beta -1 \right ) ^{2}}{ \left ( 64\,\beta -8 \right ) \left ( \left ( \left ( -1/8+\beta \right ) a{{\rm e}^{\beta \,x-x/8}}-{{\rm e}^{y\lambda }}\lambda \,b \right ){{\rm e}^{-\beta \,x-y\lambda +x/8}}+b\lambda \,{{\rm e}^{-\beta \,{\it \_a}+{\it \_a}/8}} \right ) }} \right ) ^{{\frac{\delta }{\lambda }}}{{\rm e}^{\mu \,{\it \_a}}}+k \right ) }{d{\it \_a}}}}\]

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6.4.6.10 [1069] Problem 10

problem number 1069

Added Feb. 23, 2019.

Problem Chapter 4.3.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda y} w_x + b e^{\beta x} w_y = (c e^{\mu x} + k ) w \]

Mathematica

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

Maple

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

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