100 HFOPDE, chapter 4.3.1

 100.1 Problem 1
 100.2 Problem 2
 100.3 Problem 3
 100.4 Problem 4
 100.5 Problem 5
 100.6 Problem 6
 100.7 Problem 7
 100.8 Problem 8
 100.9 Problem 9
 100.10 Problem 10

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100.1 Problem 1

problem number 852

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

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

Maple

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

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100.2 Problem 2

problem number 853

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

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

Maple

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

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100.3 Problem 3

problem number 854

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

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

Maple

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

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100.4 Problem 4

problem number 855

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

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

Maple

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

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100.5 Problem 5

problem number 856

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

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{-a\beta \,y+\lambda \,ya+b{{\rm e}^{x \left ( \beta -\lambda \right ) }}}{ \left ( \beta -\lambda \right ) a}} \right ){{\rm e}^{\int ^{x}\!{\frac{c}{a}{{\rm e}^{-{\frac{\gamma \, \left ( -a\beta \,y+\lambda \,ya+b{{\rm e}^{x \left ( \beta -\lambda \right ) }} \right ) \beta }{ \left ( \beta -\lambda \right ) ^{2}a}}+{\frac{\gamma \, \left ( -a\beta \,y+\lambda \,ya+b{{\rm e}^{x \left ( \beta -\lambda \right ) }} \right ) \lambda }{ \left ( \beta -\lambda \right ) ^{2}a}}-{\frac{\lambda \,{\it \_a}\,\beta }{\beta -\lambda }}+{\frac{{\lambda }^{2}{\it \_a}}{\beta -\lambda }}+{\frac{{{\rm e}^{{\it \_a}\, \left ( \beta -\lambda \right ) }}\gamma \,b}{ \left ( \beta -\lambda \right ) a}}}}}{d{\it \_a}}}} \]

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100.6 Problem 6

problem number 857

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

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\beta y-\lambda x} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{a \beta \lambda }\right ) \exp \left (-\frac{c \gamma e^{-\lambda x} \left (1-\frac{e^{-\beta y} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{b \beta }\right )^{\frac{\gamma }{\beta }} \left (\frac{b \beta e^{-\lambda x}}{a \lambda }-\frac{e^{-\beta y-\lambda x} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{a \lambda }\right )^{-\frac{\gamma }{\beta }} \text{Hypergeometric2F1}\left (\frac{\beta +\gamma }{\beta },\frac{\gamma }{\beta }-1,\frac{\gamma }{\beta },\frac{e^{-\beta y} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{b \beta }\right )}{a \lambda (\beta -\gamma )}+\frac{\delta s e^{-\lambda x} \left (1-\frac{e^{-\beta y} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{b \beta }\right )^{\frac{\delta }{\beta }} \left (\frac{b \beta e^{-\lambda x}}{a \lambda }-\frac{e^{-\beta y-\lambda x} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{a \lambda }\right )^{-\frac{\delta }{\beta }} \text{Hypergeometric2F1}\left (\frac{\beta +\delta }{\beta },\frac{\delta }{\beta }-1,\frac{\delta }{\beta },\frac{e^{-\beta y} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{b \beta }\right )}{a \lambda (\delta -\beta )}-\frac{c e^{-\lambda x} \left (\frac{b \beta e^{-\lambda x}}{a \lambda }-\frac{e^{-\beta y-\lambda x} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{a \lambda }\right )^{-\frac{\gamma }{\beta }}}{a \lambda }-\frac{s e^{-\lambda x} \left (\frac{b \beta e^{-\lambda x}}{a \lambda }-\frac{e^{-\beta y-\lambda x} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{a \lambda }\right )^{-\frac{\delta }{\beta }}}{a \lambda }\right )\right \}\right \} \]

Maple

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

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100.7 Problem 7

problem number 858

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

\[ \text{DSolve}\left [a e^{\beta x} w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (b e^{\gamma x}+c e^{\lambda y}\right )=w(x,y) \left (k e^{\delta y}+s e^{\mu x}+p\right ),w(x,y),\{x,y\}\right ] \]

Maple

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

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100.8 Problem 8

problem number 859

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

\[ \text{DSolve}\left [a e^{\beta x} w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (b e^{\gamma x}+c e^{\lambda y}\right )=w(x,y) \left (s e^{\delta y+\mu x}+k\right ),w(x,y),\{x,y\}\right ] \]

Maple

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

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100.9 Problem 9

problem number 860

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

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{a \lambda (\beta -\gamma )}\right ) \exp \left (-\frac{c e^{x (\mu -\beta )} \left (-\frac{a (\beta -\gamma ) e^{\beta x}}{-e^{-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )-b \lambda e^{\gamma x}}\right )^{\delta /\lambda } \left (\frac{b e^{x (\gamma -\beta )}}{\frac{\beta e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}-\frac{\gamma e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}}+1\right )^{\delta /\lambda } \text{Hypergeometric2F1}\left (\frac{\delta }{\lambda },\frac{\beta -\mu }{\beta -\gamma },\frac{\beta -\mu }{\beta -\gamma }+1,\frac{b e^{\gamma x-\beta x}}{\frac{\gamma e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}-\frac{\beta e^{-\beta x-\lambda y} \left (-a \gamma e^{\beta x}+a \beta e^{\beta x}-b \lambda e^{\gamma x+\lambda y}\right )}{\lambda (\beta -\gamma )}}\right )}{a (\beta -\mu )}-\frac{k e^{-\beta x}}{a \beta }\right )\right \}\right \} \]

Maple

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

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100.10 Problem 10

problem number 861

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

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

Maple

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