Added April 2, 2019.
Problem Chapter 5.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda x} y+ b x^n) w_y = c w + k e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y+b*x^n)*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (y e^{-\frac {a e^{\lambda x}}{\lambda }}-\int _1^xb e^{-\frac {a e^{\lambda K[1]}}{\lambda }} K[1]^ndK[1]\right )\right )}{c-\gamma }\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*x^n)*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( {\frac {k{{\rm e}^{x \left ( \gamma -c \right ) }}}{\gamma -c}}+{\it \_F1} \left ( -b\int \!{x}^{n}{{\rm e}^{-{\frac {a{{\rm e}^{x\lambda }}}{\lambda }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {a{{\rm e}^{x\lambda }}}{\lambda }}}} \right ) \right ) {{\rm e}^{cx}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda x} y+ b e^{\beta x}) w_y = c w + k e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y+b*Exp[beta*x])*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (y e^{-\frac {a e^{\lambda x}}{\lambda }}-\int _1^xb e^{\beta K[1]-\frac {a e^{\lambda K[1]}}{\lambda }}dK[1]\right )\right )}{c-\gamma }\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*exp(beta*x))*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) =-{\frac {{{\rm e}^{cx}}}{-\gamma +c} \left ( \left ( \gamma -c \right ) {\it \_F1} \left ( -b\int \!{{\rm e}^{{\frac {x\lambda \,\beta -a{{\rm e}^{x\lambda }}}{\lambda }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {a{{\rm e}^{x\lambda }}}{\lambda }}}} \right ) +k{{\rm e}^{x \left ( \gamma -c \right ) }} \right ) }\]
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Added April 2, 2019.
Problem Chapter 5.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda x} y+ b e^{\beta x}) w_y = c w + k x^n \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y+b*Exp[beta*x])*D[w[x, y], y] == c*w[x,y]+k*x^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{c x} \left (-\frac {k x^n (c x)^{-n} \text {Gamma}(n+1,c x)}{c}+c_1\left (y e^{-\frac {a e^{\lambda x}}{\lambda }}-\int _1^xb e^{\beta K[1]-\frac {a e^{\lambda K[1]}}{\lambda }}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y+b*exp(beta*x))*diff(w(x,y),y) = c*w(x,y)+k*x^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {{{\rm e}^{cx}}}{c \left ( 1+n \right ) } \left ( c \left ( 1+n \right ) {\it \_F1} \left ( -b\int \!{{\rm e}^{{\frac {x\lambda \,\beta -a{{\rm e}^{x\lambda }}}{\lambda }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {a{{\rm e}^{x\lambda }}}{\lambda }}}} \right ) +k{x}^{n} \left ( cx \right ) ^{-{\frac {n}{2}}}{{\rm e}^{-{\frac {cx}{2}}}} \WhittakerM \left ( {\frac {n}{2}},{\frac {n}{2}}+{\frac {1}{2}},cx \right ) \right ) }\]
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Added April 2, 2019.
Problem Chapter 5.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a e^{\lambda y} + b x^k) w_y = c w + k e^{\gamma x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*y]+b*x^k)*D[w[x, y], y] == c*w[x,y]+k*Exp[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {e^{c x} \left (k e^{x (\gamma -c)}+(\gamma -c) c_1\left (\frac {a \lambda x \left (-\frac {b \lambda x^{k+1}}{k+1}\right )^{-\frac {1}{k+1}} \text {Gamma}\left (\frac {1}{k+1},-\frac {b \lambda x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\lambda \left (-b x^{k+1}+k y+y\right )}{k+1}}}{a b k (k+1) \lambda ^2}\right )\right )}{c-\gamma }\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*y)+b*x^k)*diff(w(x,y),y) = c*w(x,y)+k*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) =-{\frac {{{\rm e}^{cx}}}{-\gamma +c} \left ( \left ( \gamma -c \right ) {\it \_F1} \left ( {\frac {1}{ \left ( 2\,{k}^{2}+7\,k+6 \right ) b\lambda } \left ( a \left ( -{\frac {{x}^{1+k}b\lambda }{1+k}} \right ) ^{{\frac {-k-2}{2+2\,k}}}{x}^{-k}{{\rm e}^{{\frac {{x}^{1+k}b\lambda }{2+2\,k}}}} \left ( 1+k \right ) \left ( k+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+2}{2+2\,k}},{\frac {2\,k+3}{2+2\,k}},-{\frac {{x}^{1+k}b\lambda }{1+k}} \right ) -{{\rm e}^{{\frac {{x}^{1+k}b\lambda }{2+2\,k}}}} \left ( 1+k \right ) ^{2} \left ( \left ( -k-2 \right ) {x}^{-k}+xb\lambda \right ) a \left ( -{\frac {{x}^{1+k}b\lambda }{1+k}} \right ) ^{{\frac {-k-2}{2+2\,k}}} \WhittakerM \left ( -{\frac {k}{2+2\,k}},{\frac {2\,k+3}{2+2\,k}},-{\frac {{x}^{1+k}b\lambda }{1+k}} \right ) -2\,b \left ( k+3/2 \right ) \left ( k+2 \right ) {{\rm e}^{{\frac { \left ( {x}^{1+k}b-y \left ( 1+k \right ) \right ) \lambda }{1+k}}}} \right ) } \right ) +k{{\rm e}^{x \left ( \gamma -c \right ) }} \right ) }\]
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Added April 2, 2019.
Problem Chapter 5.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x e^{\lambda x+\mu y} w + b e^{\nu x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Exp[lambda*x+mu*y]*w[x,y]+b*Exp[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {a x e^{\lambda x+\mu y}}{\lambda x+\mu y}} \left (\int _1^x\frac {b \exp \left (\nu K[1]-\frac {a e^{\left (\lambda +\frac {\mu y}{x}\right ) K[1]} x}{\lambda x+\mu y}\right )}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x* diff(w(x,y),x)+ y*diff(w(x,y),y) = a*x*exp(lambda*x+mu*y)*w(x,y)+k*exp(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{{\it \_a}}{{\rm e}^{-{\frac {1}{x\lambda +\mu \,y} \left ( ax{{\rm e}^{{\frac {\mu \,y{\it \_a}}{x}}+{\it \_a}\,\lambda }}-\nu \,{\it \_a}\, \left ( x\lambda +\mu \,y \right ) \right ) }}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) {{\rm e}^{{a{{\rm e}^{x\lambda +\mu \,y}} \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = (a y e^{\lambda x}+ b x e^{\mu y}) w + c e^{\nu x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == (a*y*Exp[lambda*x]+b*x*Exp[mu*y])*w[x,y]+c*Exp[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {a y e^{\lambda x}}{\lambda x}+\frac {b x e^{\mu y}}{\mu y}} \left (\int _1^x\frac {c \exp \left (-\frac {b e^{\frac {\mu y K[1]}{x}} x}{\mu y}+\nu K[1]-\frac {a e^{\lambda K[1]} y}{\lambda x}\right )}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x* diff(w(x,y),x)+ y*diff(w(x,y),y) = (a*y*exp(lambda*x)+b*x*exp(mu*y))*w(x,y)+c*exp(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {c}{{\it \_a}}{{\rm e}^{-{\frac {x}{\mu \,y\lambda } \left ( {\frac {{{\rm e}^{{\it \_a}\,\lambda }}{y}^{2}a\mu }{{x}^{2}}}-{\frac {\nu \,{\it \_a}\,\mu \,y\lambda }{x}}+{{\rm e}^{{\frac {\mu \,y{\it \_a}}{x}}}}b\lambda \right ) }}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) {{\rm e}^{{\frac {x}{\mu \,y\lambda } \left ( {\frac {{{\rm e}^{x\lambda }}{y}^{2}a\mu }{{x}^{2}}}+{{\rm e}^{\mu \,y}}b\lambda \right ) }}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^k w_x + b e^{\lambda x} w_y = w + c e^{\beta x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y^k*D[w[x, y], x] + b*Exp[lambda*x]*D[w[x, y], y] == w[x,y]+c*Exp[beta*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (-\frac {(k+1) y^{k+1} \left (\left (y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k} \, _2F_1\left (1,\frac {1}{k+1};\frac {k+2}{k+1};\frac {a \lambda y^{k+1}}{a \lambda y^{k+1}-b e^{\lambda x} (k+1)}\right )}{a \lambda y^{k+1}-b (k+1) e^{\lambda x}}\right ) \left (\int _1^x\frac {c \exp \left (\frac {(k+1) \left (a \lambda y^{k+1}-b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)\right ) \, _2F_1\left (1,\frac {1}{k+1};\frac {k+2}{k+1};1-\frac {b e^{\lambda K[1]} (k+1)}{b e^{\lambda x} (k+1)-a \lambda y^{k+1}}\right ) \left (\left (y^{k+1}-\frac {b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a \lambda \left (a \lambda y^{k+1}-b e^{\lambda x} (k+1)\right )}+\beta K[1]\right ) \left (\left (y^{k+1}-\frac {b \left (e^{\lambda x}-e^{\lambda K[1]}\right ) (k+1)}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b e^{\lambda x}}{a \lambda }\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*y^k* diff(w(x,y),x)+ b*exp(lambda*x)*diff(w(x,y),y) = w(x,y)+c*exp(beta*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {c}{a} \left ( \left ( {\frac {{{\rm e}^{{\it \_b}\,\lambda }}b \left ( 1+k \right ) -b{{\rm e}^{x\lambda }} \left ( 1+k \right ) +{y}^{k}ya\lambda }{a\lambda }} \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}{{\rm e}^{{\frac {1}{a} \left ( a{\it \_b}\,\beta -\int \! \left ( \left ( {\frac {{{\rm e}^{{\it \_b}\,\lambda }}b \left ( 1+k \right ) -b{{\rm e}^{x\lambda }} \left ( 1+k \right ) +{y}^{k}ya\lambda }{a\lambda }} \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}\,{\rm d}{\it \_b} \right ) }}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {-b{{\rm e}^{x\lambda }} \left ( 1+k \right ) +{y}^{k}ya\lambda }{a\lambda }} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( \left ( {\frac {{{\rm e}^{{\it \_a}\,\lambda }}b \left ( 1+k \right ) -b{{\rm e}^{x\lambda }} \left ( 1+k \right ) +{y}^{k}ya\lambda }{a\lambda }} \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda x} w_x + b y w_y = w + c e^{\lambda x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*y*D[w[x, y], y] == w[x,y]+c*Exp[lambda*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {e^{-\frac {e^{-\lambda x}}{a \lambda }} \left (-c \text {Ei}\left (\frac {e^{-\lambda x}}{a \lambda }\right )+a \lambda c_1\left (y e^{\frac {b e^{-\lambda x}}{a \lambda }}\right )\right )}{a \lambda }\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*x)* diff(w(x,y),x)+ b*y*diff(w(x,y),y) = w(x,y)+c*exp(lambda*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( y{{\rm e}^{{\frac {b{{\rm e}^{-x\lambda }}}{a\lambda }}}} \right ) a\lambda +c\Ei \left ( 1,-{\frac {{{\rm e}^{-x\lambda }}}{a\lambda }} \right ) \right ) {{\rm e}^{-{\frac {{{\rm e}^{-x\lambda }}}{a\lambda }}}}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda y} w_x + b x^k w_y = w + c e^{\beta x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*y]*D[w[x, y], x] + b*x^k*D[w[x, y], y] == w[x,y]+c*Exp[beta*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {(k+1) x \, _2F_1\left (1,\frac {1}{k+1};1+\frac {1}{k+1};\frac {b \lambda x^{k+1}}{b \lambda x^{k+1}-a e^{\lambda y} (k+1)}\right )}{a (k+1) e^{\lambda y}-b \lambda x^{k+1}}\right ) \left (\int _1^x\frac {c \exp \left (\left (\beta -\frac {(k+1) \, _2F_1\left (1,\frac {1}{k+1};1+\frac {1}{k+1};\frac {b \lambda K[1]^{k+1}}{b \lambda x^{k+1}-a e^{\lambda y} (k+1)}\right )}{a e^{\lambda y} (k+1)-b \lambda x^{k+1}}\right ) K[1]\right ) (k+1)}{a e^{\lambda y} (k+1)+b \lambda \left (K[1]^{k+1}-x^{k+1}\right )}dK[1]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b x^{k+1}}{a k+a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*y)* diff(w(x,y),x)+ b*x^k*diff(w(x,y),y) = w(x,y)+c*exp(beta*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {c \left ( 1+k \right ) }{-{x}^{1+k}b\lambda +b\lambda \,{{\it \_b}}^{1+k}+{{\rm e}^{\lambda \,y}}a \left ( 1+k \right ) }{{\rm e}^{{\frac {1}{b\lambda } \left ( \left ( -k-1 \right ) \int \!{\frac {b\lambda }{-{x}^{1+k}b\lambda +b\lambda \,{{\it \_b}}^{1+k}+{{\rm e}^{\lambda \,y}}a \left ( 1+k \right ) }}\,{\rm d}{\it \_b}+b\beta \,\lambda \,{\it \_b} \right ) }}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {-{x}^{1+k}b\lambda +{{\rm e}^{\lambda \,y}}a \left ( 1+k \right ) }{ \left ( 1+k \right ) b\lambda }} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1+k}{-{x}^{1+k}b\lambda +b\lambda \,{{\it \_a}}^{1+k}+{{\rm e}^{\lambda \,y}}a \left ( 1+k \right ) }}{d{\it \_a}}}}\]
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Added April 2, 2019.
Problem Chapter 5.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda y} w_x + b e^{\beta x} w_y = w + c x^k \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*y]*D[w[x, y], x] + b*Exp[beta*x]*D[w[x, y], y] == w[x,y]+c*x^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\beta x-\log \left (\frac {a \beta e^{\lambda y}}{\lambda }\right )}{a \beta e^{\lambda y}-b \lambda e^{\beta x}}\right ) \left (\int _1^x\frac {\beta c \exp \left (\frac {\log \left (\frac {a e^{\lambda y} \beta }{\lambda }+b \left (-e^{\beta x}+e^{\beta K[1]}\right )\right )-\beta K[1]}{a \beta e^{\lambda y}-b e^{\beta x} \lambda }\right ) K[1]^k}{a e^{\lambda y} \beta +b \left (-e^{\beta x}+e^{\beta K[1]}\right ) \lambda }dK[1]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\frac {b e^{\beta x}}{a \beta }\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*y)* diff(w(x,y),x)+ b*exp(beta*x)*diff(w(x,y),y) = w(x,y)+c*x^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\it \_a}}^{k}c\beta \, \left ( {{\rm e}^{\beta \,{\it \_a}}} \right ) ^{ \left ( {{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{\lambda \,y}}a\beta \right ) ^{-1}}}{b\lambda } \left ( -{\frac {{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{\lambda \,y}}a\beta }{b\lambda }}+{{\rm e}^{\beta \,{\it \_a}}} \right ) ^{{\frac {-{{\rm e}^{\beta \,x}}b\lambda +{{\rm e}^{\lambda \,y}}a\beta -1}{{{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{\lambda \,y}}a\beta }}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {-{{\rm e}^{\beta \,x}}b\lambda +{{\rm e}^{\lambda \,y}}a\beta }{b\beta \,\lambda }} \right ) \right ) \left ( {{\rm e}^{\beta \,x}} \right ) ^{ \left ( -{{\rm e}^{\beta \,x}}b\lambda +{{\rm e}^{\lambda \,y}}a\beta \right ) ^{-1}} \left ( {\frac {{{\rm e}^{\lambda \,y}}a\beta }{b\lambda }} \right ) ^{ \left ( {{\rm e}^{\beta \,x}}b\lambda -{{\rm e}^{\lambda \,y}}a\beta \right ) ^{-1}}\]
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