Added January 7, 2019.
Problem 2.3.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2+a \lambda e^{\lambda x}- a^2 e^{2 \lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*lambda*Exp[lambda*x] - a^2*Exp[2*lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\text {Ei}\left (\frac {2 a e^{\lambda x}}{\lambda }\right ) \left (y-a e^{\lambda x}\right )+\lambda e^{\frac {2 a e^{\lambda x}}{\lambda }}}{a e^{\lambda x}-y}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (y^2+a*lambda*exp(lambda*x)- a^2*exp(2*lambda *x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+y}{\lambda \,{\mathrm e}^{\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }}+\left (a \,{\mathrm e}^{\lambda x}-y \right ) \Ei \left (1, -\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right )}\right )\]
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Added January 7, 2019.
Problem 2.3.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2+b y+ a (\lambda -b) e^{\lambda x} - a^2 e^{2 \lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + b*y + a*(lambda - b)*Exp[lambda*x] - a^2*Exp[2*lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2^{b/\lambda } \lambda ^{-\frac {b}{\lambda }} e^{b x} a^{b/\lambda } \left (\left (a \left (-e^{\lambda x}\right )+b+y\right ) \operatorname {LaguerreL}\left (-\frac {b}{\lambda },\frac {b}{\lambda },\frac {2 a e^{\lambda x}}{\lambda }\right )-2 a e^{\lambda x} \operatorname {LaguerreL}\left (-\frac {b+\lambda }{\lambda },\frac {b+\lambda }{\lambda },\frac {2 a e^{\lambda x}}{\lambda }\right )\right )}{a e^{\lambda x}-y}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (y^2+b*y+ a*(lambda-b)*exp(lambda*x) - a^2*exp(2*lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (-a \,{\mathrm e}^{\lambda x}+y \right ) \left (\int {\mathrm e}^{\frac {b \lambda x +2 a \,{\mathrm e}^{\lambda x}}{\lambda }}d x \right )+{\mathrm e}^{\frac {b \lambda x +2 a \,{\mathrm e}^{\lambda x}}{\lambda }}}{a \,{\mathrm e}^{\lambda x}-y}\right )\]
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Added January 7, 2019.
Problem 2.3.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2+a e^{\lambda x} y-a b e^{\lambda x} - b^2 \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*Exp[lambda*x]*y - a*b*Exp[lambda*x] - b^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (2 b (-1)^{-\frac {b}{\lambda }} \left (-\frac {\operatorname {Gamma}\left (\frac {2 b}{\lambda },0,-\frac {a e^{\lambda x}}{\lambda }\right )}{\lambda }+\frac {\lambda ^{-\frac {2 b}{\lambda }} a^{\frac {2 b}{\lambda }} e^{\frac {a e^{\lambda x}+2 b \lambda x+2 i \pi b}{\lambda }}}{b-y}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (y^2+a*exp(lambda*x)*y-a*b*exp(lambda*x)- b^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (b -y \right ) \left (\int {\mathrm e}^{\frac {2 b \lambda x +a \,{\mathrm e}^{\lambda x}}{\lambda }}d x \right )-{\mathrm e}^{\frac {2 b \lambda x +a \,{\mathrm e}^{\lambda x}}{\lambda }}}{b -y}\right )\]
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Added January 7, 2019.
Problem 2.3.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x - \left ( y^2-a x e^{\lambda x} y + a e^{\lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] - (y^2 - a*x*Exp[lambda*x]*y + a*Exp[lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {a e^{\lambda x} (\lambda x-1)}{\lambda ^2}}}{x (x y-1)}-\int _1^x\frac {e^{\frac {a e^{\lambda K[1]} (\lambda K[1]-1)}{\lambda ^2}}}{K[1]^2}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)- (y^2-a*x*exp(lambda*x)*y + a*exp(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (y x^{2}-x \right ) \left (\int \frac {{\mathrm e}^{\frac {\left (\lambda x -1\right ) a \,{\mathrm e}^{\lambda x}}{\lambda ^{2}}}}{x^{2}}d x \right )-{\mathrm e}^{\frac {\left (\lambda x -1\right ) a \,{\mathrm e}^{\lambda x}}{\lambda ^{2}}}}{\left (x y -1\right ) \lambda ^{2} x}\right )\]
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Added January 7, 2019.
Problem 2.3.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left (a e^{\lambda x} y^2 + b e^{-\lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^2 + b*Exp[-(lambda*x)])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{x \left (-\sqrt {\lambda ^2-4 a b}\right )} \left (\sqrt {\lambda ^2-4 a b}-2 a y e^{\lambda x}-\lambda \right )}{a \left (2 y e^{\lambda x} \sqrt {\lambda ^2-4 a b}-4 b\right )+\lambda \left (\sqrt {\lambda ^2-4 a b}+\lambda \right )}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y^2 + b*exp(-lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (2 \lambda \arctan \left (\frac {2 a \lambda y \,{\mathrm e}^{\lambda x}+\lambda ^{2}}{\sqrt {4 \lambda ^{2} a b -\lambda ^{4}}}\right )-\sqrt {4 \lambda ^{2} a b -\lambda ^{4}}\, x \right ) \lambda }{\sqrt {4 \lambda ^{2} a b -\lambda ^{4}}}\right )\]
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Added January 7, 2019.
Problem 2.3.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left (a e^{\lambda x} y^2 + b \mu e^{\mu x} - a b^2 e^{(\lambda + 2 \mu )x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^2 + b*mu*Exp[mu*x] - a*b^2*Exp[(lambda + 2*mu)*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y^2 + b*mu*exp(mu*x) - a*b^2*exp((lambda + 2*mu)*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 7, 2019.
Problem 2.3.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left (a e^{\lambda x} y^2 + b y + c e^{-\lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^2 + b*y + c*Exp[-(lambda*x)])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{x \left (-\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}\right )} \left (\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}-2 a y e^{\lambda x}-b-\lambda \right )}{a \left (2 y e^{\lambda x} \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}-4 c\right )+b \left (\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+2 \lambda \right )+\lambda \left (\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+\lambda \right )+b^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y^2 + b*y +c*exp(-lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (b +\lambda \right ) \left (-\frac {\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}\, x}{2}+\left (b +\lambda \right ) \arctan \left (\frac {\left (b +\lambda \right ) \left (2 a y \,{\mathrm e}^{\lambda x}+b +\lambda \right )}{\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}}\right )\right )}{\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}}\right )\]
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Added January 7, 2019.
Problem 2.3.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left (a e^{\lambda x} y^2 + \mu y - a b^2 e^{(\lambda +2 \mu )x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^2 + mu*y - a*b^2*Exp[(lambda + 2*mu)*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y^2 + mu*y - a*b^2*exp((lambda+2*mu)*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-b \cosh \left (\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right ) {\mathrm e}^{\left (\lambda +\mu \right ) x}-y \,{\mathrm e}^{\lambda x} \sinh \left (\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right )}{b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} \sinh \left (\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right )+y \cosh \left (\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right ) {\mathrm e}^{\lambda x}}\right )\]
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Added January 7, 2019.
Problem 2.3.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left (e^{\lambda x} y^2 + a e^{\mu x} y+a \lambda e^{(\mu -lambda)x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (Exp[lambda*x]*y^2 + a*Exp[mu*x]*y + a*lambda*Exp[(mu - lambda)*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left ((-1)^{\lambda /\mu } \mu ^{-\frac {\lambda }{\mu }} a^{\lambda /\mu } \operatorname {Gamma}\left (-\frac {\lambda }{\mu },-\frac {a e^{\mu x}}{\mu }\right )-\frac {\mu e^{\frac {a e^{\mu x}}{\mu }-\lambda x}}{y e^{\lambda x}+\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (exp(lambda*x)*y^2 + a*exp(mu*x)*y+a*lambda*exp((mu-lambda)*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (-\lambda +\mu \right ) \left (y \,{\mathrm e}^{\lambda x}+\lambda \right ) {\mathrm e}^{\lambda x}}{a \lambda \hypergeom \left (\left [\frac {-\lambda +\mu }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) {\mathrm e}^{\mu x}-\left (-\lambda +\mu \right ) y \hypergeom \left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {-\lambda +\mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) {\mathrm e}^{\lambda x}}\right )\]
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Added January 7, 2019.
Problem 2.3.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x -\left ( \lambda e^{\lambda x} y^2 - a e^{\mu x} y+a e^{(\mu -lambda)x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] - (lambda*Exp[lambda*x]*y^2 - a*Exp[mu*x]*y + a*lambda*Exp[(mu - lambda)*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\mu \left (a e^{\mu x} \operatorname {LaguerreL}\left (-\frac {-\lambda ^2+\lambda +\mu }{\mu },\frac {\lambda +\mu }{\mu },\frac {a e^{\mu x}}{\mu }\right )+\lambda \left (y e^{\lambda x}-1\right ) \operatorname {LaguerreL}\left (\frac {(\lambda -1) \lambda }{\mu },\frac {\lambda }{\mu },\frac {a e^{\mu x}}{\mu }\right )\right )}{\lambda \left (a (\lambda -1) e^{\mu x} \operatorname {HypergeometricU}\left (\frac {-\lambda ^2+\lambda +\mu }{\mu },\frac {\lambda }{\mu }+2,\frac {a e^{\mu x}}{\mu }\right )+\left (\mu -\mu y e^{\lambda x}\right ) \operatorname {HypergeometricU}\left (-\frac {(\lambda -1) \lambda }{\mu },\frac {\lambda +\mu }{\mu },\frac {a e^{\mu x}}{\mu }\right )\right )}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)- (lambda*exp(lambda*x)*y^2 - a*exp(mu*x)*y + a*lambda*exp((mu-lambda)*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (-\lambda ^{2}-\mu \right ) M\left (\frac {-\lambda ^{2}+\lambda -\mu }{\mu }, \frac {\lambda +\mu }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )-\left (-\lambda y \,{\mathrm e}^{\lambda x}+a \,{\mathrm e}^{\mu x}-\lambda ^{2}+\lambda -\mu \right ) M\left (-\frac {\left (\lambda -1\right ) \lambda }{\mu }, \frac {\lambda +\mu }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )}{-\mu U\left (\frac {-\lambda ^{2}+\lambda -\mu }{\mu }, \frac {\lambda +\mu }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+\left (-\lambda y \,{\mathrm e}^{\lambda x}+a \,{\mathrm e}^{\mu x}-\lambda ^{2}+\lambda -\mu \right ) U\left (-\frac {\left (\lambda -1\right ) \lambda }{\mu }, \frac {\lambda +\mu }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )}\right )\]
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Added January 7, 2019.
Problem 2.3.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{\lambda x} y^2+ a b e^{(\lambda + \mu )x} y - b \mu e^{\mu x}\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^2 + a*b*Exp[(lambda + mu)*x]*y - b*mu*Exp[mu*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp(lambda*x)*y^2+ a*b*exp((lambda +mu)*x)*y - b*mu*exp(mu*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+y \,{\mathrm e}^{\lambda x}\right ) a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }}}{2 \left (\frac {3 \lambda }{2}+\mu \right ) \left (2 \lambda +\mu \right )^{2} \WhittakerM \left (\frac {4 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {3 \lambda +2 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-3 \left (\lambda +\mu \right ) \left (\frac {2 \lambda }{3}+\mu \right ) x}{2 \lambda +2 \mu }}+2 \left (\left (\left (\frac {3 \lambda }{2}+\mu \right ) b \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\left (\lambda +\mu \right ) \mu x}{2 \lambda +2 \mu }}+\frac {\left (2 \lambda +\mu \right ) y \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-3 \left (\lambda +\mu \right ) \mu x}{2 \lambda +2 \mu }}}{2}\right ) a -\frac {\left (2 \lambda +\mu \right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-3 \left (\lambda +\mu \right ) \left (\frac {2 \lambda }{3}+\mu \right ) x}{2 \lambda +2 \mu }}}{2}\right ) \left (2 \lambda +\mu \right ) \WhittakerM \left (\frac {2 \lambda +\mu }{2 \lambda +2 \mu }, \frac {3 \lambda +2 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right )+\left (\left (a b^{2} {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+\left (2 \lambda +\mu \right ) \left (\lambda +\mu \right ) x}{2 \lambda +2 \mu }}+a b y \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\left (\lambda +\mu \right ) \left (-2 \lambda +\mu \right ) x}{2 \lambda +2 \mu }}+\left (2 \lambda +\mu \right ) \left (b \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\left (\lambda +\mu \right ) \mu x}{2 \lambda +2 \mu }}+y \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-3 \left (\lambda +\mu \right ) \mu x}{2 \lambda +2 \mu }}\right )\right ) a -\left (2 \lambda +\mu \right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-3 \left (\lambda +\mu \right ) \left (\frac {2 \lambda }{3}+\mu \right ) x}{2 \lambda +2 \mu }}\right ) \left (\lambda +\mu \right ) \WhittakerM \left (-\frac {\mu }{2 \lambda +2 \mu }, \frac {3 \lambda +2 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right )}\right )\]
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Added January 7, 2019.
Problem 2.3.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{(2 \lambda + \mu ) x} y^2+ \left (b e^{(\lambda + \mu )x} -\lambda \right ) y + c e^{\mu x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[(2*lambda + mu)*x]*y^2 + (b*Exp[(lambda + mu)*x] - lambda)*y + c*Exp[mu*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {i \pi e^{-\frac {\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}} \left (\sqrt {b^2-4 a c}-2 a y e^{\lambda x}-b\right )}{2 \left (\left (2 a y e^{\lambda x}+b\right ) \cosh \left (\frac {\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )+\sqrt {b^2-4 a c} \sinh \left (\frac {\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )\right )}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*exp((2*lambda +mu)*x)*y^2+ (b*exp((lambda +mu)*x) -lambda)*y + c*exp(mu*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (-2 \left (\lambda +\mu \right ) b \arctan \left (\frac {2 a b y \,{\mathrm e}^{\lambda x}+b^{2}}{\sqrt {4 a c b^{2}-b^{4}}}\right )+\sqrt {4 a c b^{2}-b^{4}}\, {\mathrm e}^{\left (\lambda +\mu \right ) x}\right ) b}{\sqrt {4 a c b^{2}-b^{4}}\, \left (\lambda +\mu \right )}\right )\]
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Added January 7, 2019.
Problem 2.3.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( e^{\lambda x} \left ( y- b e^{\mu x} \right )^2 + b \mu e^{\mu x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (Exp[lambda*x]*(y - b*Exp[mu*x])^2 + b*mu*Exp[mu*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {b \left (-e^{x (\lambda +\mu )}\right )+y e^{\lambda x}+\lambda }{\lambda \left (b e^{\mu x}-y\right )}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ ( exp(lambda*x) *(y- b*exp(mu*x))^2 + b*mu*exp(mu*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-y \,{\mathrm e}^{\lambda x}-\lambda }{\left (b \,{\mathrm e}^{\mu x}-y \right ) \lambda }\right )\]
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Added January 7, 2019.
Problem 2.3.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{\lambda x} y^2+ b n x^{n-1} - a b^2 e^{\lambda x} x^{2 n} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^2 + b*n*x^(n - 1) - a*b^2*Exp[lambda*x]*x^(2*n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+ ( a*exp(lambda*x)*y^2+ b*n*x^(n-1) - a*b^2*exp(lambda*x)*x^(2*n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 7, 2019.
Problem 2.3.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( e^{\lambda x} y^2+ a x^n y + a \lambda x^n e^{-\lambda x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (Exp[lambda*x]*y^2 + a*x^n*y + a*lambda*x^n*Exp[-(lambda*x)])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+ ( exp(lambda*x)*y^2+ a*x^n*y + a*lambda*x^n*exp(-lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (-y \,{\mathrm e}^{\lambda x}-\lambda \right ) \left (\int {\mathrm e}^{\frac {\left (a x^{n}-\left (n +1\right ) \lambda \right ) x}{n +1}}d x \right )-{\mathrm e}^{\frac {\left (a x^{n}-\left (n +1\right ) \lambda \right ) x}{n +1}}}{y \,{\mathrm e}^{\lambda x}+\lambda }\right )\]
____________________________________________________________________________________
Added January 7, 2019.
Problem 2.3.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( \lambda e^{\lambda x} y^2+ a x^n e^{\lambda x} y - a x^n e^{2 \lambda x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*Exp[lambda*x]*y^2 + a*x^n*Exp[lambda*x]*y - a*x^n*Exp[2*lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+ ( lambda*exp(lambda*x)*y^2+ a*x^n*exp(lambda*x)*y - a*x^n*exp(2*lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 7, 2019.
Problem 2.3.2.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{\lambda x} y^2- a b x^n e^{\lambda x} y + b n x^{n-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^2 - a*b*x^n*Exp[lambda*x]*y + b*n*x^(n - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+ ( a*exp(lambda*x)*y^2- a*b*x^n*exp(lambda*x)*y + b*n*x^(n-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (b x^{n}-y \right ) a}{\left (b x^{n}-y \right ) a \left (\int \lambda \,{\mathrm e}^{\frac {\left (-\Gamma \left (n \right )+\Gamma \left (n , -\lambda x \right )\right ) a b n x^{n} \left (-\lambda \right )^{n} \left (-\lambda \right )^{-n} \left (-\lambda x \right )^{-n}+a b x^{n} \left (-\lambda \right )^{n} \left (-\lambda \right )^{-n} {\mathrm e}^{\lambda x}+\lambda ^{2} x}{\lambda }}d x \right )-\lambda \,{\mathrm e}^{\frac {\left (\left (-\Gamma \left (n \right )+\Gamma \left (n , -\lambda x \right )\right ) n \left (-\lambda x \right )^{-n}+{\mathrm e}^{\lambda x}\right ) a b x^{n} \left (-\lambda \right )^{n} \left (-\lambda \right )^{-n}}{\lambda }}}\right )\]
____________________________________________________________________________________
Added January 7, 2019.
Problem 2.3.2.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a x^n y^2 + b \lambda e^{\lambda x} - a b^2 x^n e^{2 \lambda x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*lambda*Exp[lambda*x] - a*b^2*x^n*Exp[2*lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+ ( a*x^n*y^2 + b*lambda*exp(lambda*x) - a*b^2*x^n*exp(2*lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 7, 2019.
Problem 2.3.2.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a x^n y^2 + \lambda y - a b^2 x^n e^{2 \lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 + lambda*y - a*b^2*x^n*Exp[2*lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-i \left (a b (-1)^{-n} \lambda ^{-n-1} \operatorname {Gamma}(n+1,-\lambda x)+\tanh ^{-1}\left (\frac {y e^{-\lambda x}}{b}\right )\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ ( a*x^n*y^2 + lambda*y - a*b^2*x^n*exp(2*lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-i \left (\left (n \Gamma \left (n , -\lambda x \right )-\Gamma \left (n +1\right )\right ) \left (-\lambda x \right )^{-n}+{\mathrm e}^{\lambda x}\right ) a b x^{n}-i \lambda \arctanh \left (\frac {y \,{\mathrm e}^{-\lambda x}}{b}\right )}{\lambda }\right )\]
____________________________________________________________________________________
Added January 7, 2019.
Problem 2.3.2.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a x^n y^2 - a b x^n e^{\lambda x} y + b \lambda e^{\lambda x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 - a*b*x^n*Exp[lambda*x]*y + b*lambda*Exp[lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+ ( a*x^n*y^2 - a*b*x^n*exp(lambda*x)*y + b*lambda*exp(lambda*x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 7, 2019.
Problem 2.3.2.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a x^n y^2 - a x^n \left (b e^{\lambda x} + c \right )y + b \lambda e^{\lambda x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 - a*x^n*(b*Exp[lambda*x] + c)*y + b*lambda*Exp[lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+ ( a*x^n*y^2 - a*x^n*(b*exp(lambda*x) + c )*y + b*lambda*exp(lambda*x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 7, 2019.
Problem 2.3.2.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a x^n e^{2 \lambda x} y^2 + \left ( b x^n e^{\lambda x} - \lambda \right ) y + c x^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*Exp[2*lambda*x]*y^2 + (b*x^n*Exp[lambda*x] - lambda)*y + c*x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\sqrt {a} \sqrt {c} \left ((-1)^{1-n} \lambda ^{-n-1} \operatorname {Gamma}(n+1,-\lambda x)-\frac {2 \tan ^{-1}\left (\frac {b-2 a y e^{\lambda x}}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*x^n*exp(2*lambda*x)*y^2 + (b*x^n*exp(lambda*x) - lambda)*y + c*x^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (b \lambda \arctan \left (\frac {2 a b y \,{\mathrm e}^{\lambda x}+b^{2}}{\sqrt {4 a c b^{2}-b^{4}}}\right )-\frac {\sqrt {4 a c b^{2}-b^{4}}\, \left (\left (n \Gamma \left (n , -\lambda x \right )-\Gamma \left (n +1\right )\right ) \left (-\lambda x \right )^{-n}+{\mathrm e}^{\lambda x}\right ) x^{n}}{2}\right ) b}{\sqrt {4 a c b^{2}-b^{4}}\, \lambda }\right )\]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{\lambda x} (y- b x^n - c)^2 +b n x^{n-1} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*(y - b*x^n - c)^2 + b*n*x^(n - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {a e^{\lambda x}}{\lambda }-\frac {1}{b x^n+c-y}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ ( a*exp(lambda*x)*(y- b*x^n - c)^2 +b*n*x^(n-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (b x^{n}+c -y \right ) a \,{\mathrm e}^{\lambda x}-\lambda }{\left (b x^{n}+c -y \right ) \lambda }\right )\]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( y^2+2 a \lambda x e^{\lambda x^2} - a^2 e^{2\lambda x^2}\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + 2*a*lambda*x*Exp[lambda*x^2] - a^2*Exp[2*lambda*x^2])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+ ( y^2+2*a*lambda*x*exp(lambda*x^2) - a^2*exp(2*lambda*x^2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{-\lambda x^2} y^2 + \lambda x y + a b^2 \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[-(lambda*x^2)]*y^2 + lambda*x*y + a*b^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y e^{-\frac {\lambda x^2}{2}}}{b}\right )-\frac {\sqrt {\frac {\pi }{2}} a b \text {erf}\left (\frac {\sqrt {\lambda } x}{\sqrt {2}}\right )}{\sqrt {\lambda }}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ ( a*exp(-lambda*x^2)*y^2 + lambda*x*y + a*b^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\sqrt {\pi }\, \sqrt {2}\, a b \erf \left (\frac {\sqrt {2}\, \sqrt {\lambda }\, x}{2}\right )-2 \sqrt {\lambda }\, \arctan \left (\frac {y \,{\mathrm e}^{-\frac {\lambda x^{2}}{2}}}{b}\right )}{2 \sqrt {\lambda }}\right )\]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a x^n y^2 + \lambda x y + a b^2 x^n e^{\lambda x^2}\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 + lambda*x*y + a*b^2*x^n*Exp[lambda*x^2])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y e^{-\frac {\lambda x^2}{2}}}{b}\right )-i a b i^{-n} 2^{\frac {n-1}{2}} \lambda ^{-\frac {n}{2}-\frac {1}{2}} \operatorname {Gamma}\left (\frac {n+1}{2},-\frac {\lambda x^2}{2}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ ( a*x^n*y^2 + lambda*x*y + a*b^2*x^n*exp(lambda*x^2) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (-\lambda x^{2}\right )^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}\right )-a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (-\lambda x^{2}\right )^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}, -\frac {\lambda x^{2}}{2}\right )-\arctan \left (\frac {y \,{\mathrm e}^{-\frac {\lambda x^{2}}{2}}}{b}\right )\right )\]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{2 \lambda x} y^3 + b e^{\lambda x} y^2 + c y+ d e^{-\lambda x}\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[2*lambda*x]*y^3 + b*Exp[lambda*x]*y^2 + c*y + d*Exp[-(lambda*x)])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+ ( a*exp(2*lambda*x)*y^3 + b*exp(lambda*x)*y^2 + c*y+ d*exp(-lambda*x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (x -\frac {\ln \left (y \,{\mathrm e}^{\lambda x}-\RootOf \left (\mathit {\_Z}^{3} a +\mathit {\_Z}^{2} b +\left (c +\lambda \right ) \mathit {\_Z} +d \right )\right )}{3 \RootOf \left (\mathit {\_Z}^{3} a +\mathit {\_Z}^{2} b +\left (c +\lambda \right ) \mathit {\_Z} +d \right )^{2} a +2 \RootOf \left (\mathit {\_Z}^{3} a +\mathit {\_Z}^{2} b +\left (c +\lambda \right ) \mathit {\_Z} +d \right ) b +c +\lambda }\right )\] Solution contains RootOf
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a e^{\lambda x} y^3 + 3 a b e^{\lambda x} y^2 + c y- 2 a b^3 e^{\lambda x} + b c\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Exp[lambda*x]*y^3 + 3*a*b*Exp[lambda*x]*y^2 + c*y - 2*a*b^3*Exp[lambda*x] + b*c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {6 a b^2 e^{\lambda x}}{\lambda }} \left (2 (b+y)^2 e^{\frac {6 a b^2 e^{\lambda x}}{\lambda }} \int _1^xa e^{(2 c+\lambda ) K[1]-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }}dK[1]+e^{2 c x}\right )}{(b+y)^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ ( a*exp(lambda*x)*y^3 + 3*a*b*exp(lambda*x)*y^2 + c*y- 2*a*b^3*exp(lambda*x) + b*c )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (b +y \right )^{2} a \left (\int {\mathrm e}^{\frac {-6 a b^{2} {\mathrm e}^{\lambda x}+2 \left (c +\frac {\lambda }{2}\right ) \lambda x}{\lambda }}d x \right )+{\mathrm e}^{-\frac {6 a b^{2} {\mathrm e}^{\lambda x}}{\lambda }+2 c x}}{\left (b +y \right )^{2}}\right )\]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x +\left ( a e^{\lambda x} y^2 + k y + a b^2 x^{2 k} e^{\lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*Exp[lambda*x]*y^2 + k*y + a*b^2*x^(2*k)*Exp[lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (a \sqrt {b^2} x^k (-\lambda x)^{-k} \operatorname {Gamma}(k,-\lambda x)+\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ ( a*exp(lambda*x)* y^2 + k*y + a*b^2*x^(2*k)*exp(lambda*x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (a b x^{k} \left (-\lambda x \right )^{-k} \Gamma \left (k \right )-a b x^{k} \left (-\lambda x \right )^{-k} \Gamma \left (k , -\lambda x \right )-\arctan \left (\frac {y x^{-k}}{b}\right )\right )\]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.3.2.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x +\left ( a x^{2 n} e^{\lambda x} y^2 + (b x^n e^{\lambda x} - n) y + c e^{\lambda x} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*x^(2*n)*Exp[lambda*x]*y^2 + (b*x^n*Exp[lambda*x] - n)*y + c*Exp[lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (c (-\lambda x)^{-n} \sqrt {\frac {a x^{2 n}}{c}} \operatorname {Gamma}(n,-\lambda x)-\frac {2 \sqrt {a} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \left (\sqrt {\frac {b^2}{a c}}-2 y \sqrt {\frac {a x^{2 n}}{c}}\right )}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ ( a*x^(2*n)*exp(lambda*x)*y^2 + (b*x^n*exp(lambda*x) - n)*y + c*exp(lambda*x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (b \arctan \left (\frac {2 a b y x^{n}+b^{2}}{\sqrt {4 a c b^{2}-b^{4}}}\right )+\frac {\sqrt {4 a c b^{2}-b^{4}}\, \left (-\Gamma \left (n \right )+\Gamma \left (n , -\lambda x \right )\right ) x^{n} \left (-\lambda x \right )^{-n}}{2}\right ) b}{\sqrt {4 a c b^{2}-b^{4}}}\right )\]
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Added January 10, 2019.
Problem 2.3.2.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ y w_x + e^{\lambda x} \left ( (2 a \lambda x+a + b)y - e^{\lambda x}(a^2 \lambda x^2 + a b x -c) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = y*D[w[x, y], x] + Exp[lambda*x]*((2*a*lambda*x + a + b)*y - Exp[lambda*x]*(a^2*lambda*x^2 + a*b*x - c))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := y*diff(w(x,y),x)+ exp(lambda*x)* ( (2*a*lambda*x+a + b)*y - exp(lambda*x)*(a^2*lambda*x^2 + a*b*x-c) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-\sqrt {\frac {-b^{2}-4 \lambda c}{a^{2}}}\, a \left (\int _{}^{\frac {2 \arctan \left (\frac {2 a \lambda x +b -2 \lambda y \,{\mathrm e}^{-\lambda x}}{a \sqrt {\frac {-b^{2}-4 \lambda c}{a^{2}}}}\right )}{\sqrt {\frac {-b^{2}-4 \lambda c}{a^{2}}}}}{\mathrm e}^{-\mathit {\_a}} \tan \left (\frac {\sqrt {\frac {-b^{2}-4 \lambda c}{a^{2}}}\, \mathit {\_a}}{2}\right )d\mathit {\_a} \right )-\left (2 a \lambda x +b \right ) {\mathrm e}^{-\frac {2 \arctan \left (\frac {2 a \lambda x +b -2 \lambda y \,{\mathrm e}^{-\lambda x}}{a \sqrt {\frac {-b^{2}-4 \lambda c}{a^{2}}}}\right )}{\sqrt {\frac {-b^{2}-4 \lambda c}{a^{2}}}}}}{2 a}\right )\]
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Added January 10, 2019.
Problem 2.3.2.32 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b y^m w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*y^m*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{-\lambda x}}{a \lambda }-\frac {y^{1-m}}{m-1}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*exp(lambda*x)*diff(w(x,y),x)+ b*y^m*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a \lambda y^{-m +1}-\left (m -1\right ) b \,{\mathrm e}^{-\lambda x}}{a \lambda }\right )\]
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Added January 10, 2019.
Problem 2.3.2.33 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a e^y + b x) w_x + w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*Exp[y] + b*x)*D[w[x, y], x] + D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*exp(y)+b*x)*diff(w(x,y),x)+ diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (a \,{\mathrm e}^{b y}+\left (b -1\right ) x \,{\mathrm e}^{\left (b -1\right ) y}\right ) {\mathrm e}^{\left (-2 b +1\right ) y}}{b -1}\right )\]
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Added January 10, 2019.
Problem 2.3.2.34 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n e^{\lambda y} + b x y^m) w_x + e^{\mu y} w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*Exp[lambda*y] + b*x*y^m)*D[w[x, y], x] + Exp[mu*y]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n*exp(lambda*y)+ b*x*y^m)*diff(w(x,y),x)+ exp(mu*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (x^{\frac {1}{m +1}} x^{\frac {m}{m +1}} x^{-\frac {n}{m +1}} x^{-\frac {m n}{m +1}} {\mathrm e}^{\frac {b n y^{m} \left (\mu y \right )^{-\frac {m}{2}} \WhittakerM \left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) {\mathrm e}^{-\frac {\mu y}{2}}}{\left (m +1\right ) \mu }} {\mathrm e}^{-\frac {b y^{m} \left (\mu y \right )^{-\frac {m}{2}} \WhittakerM \left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) {\mathrm e}^{-\frac {\mu y}{2}}}{\left (m +1\right ) \mu }}+a n \left (\int {\mathrm e}^{\frac {\left (n -1\right ) b y^{m} \left (\mu y \right )^{-\frac {m}{2}} \WhittakerM \left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) {\mathrm e}^{-\frac {\mu y}{2}}-\left (-\lambda +\mu \right ) \left (m +1\right ) \mu y}{\left (m +1\right ) \mu }}d y \right )-a \left (\int {\mathrm e}^{\frac {\left (n -1\right ) b y^{m} \left (\mu y \right )^{-\frac {m}{2}} \WhittakerM \left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) {\mathrm e}^{-\frac {\mu y}{2}}-\left (-\lambda +\mu \right ) \left (m +1\right ) \mu y}{\left (m +1\right ) \mu }}d y \right )\right )\]
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Added January 10, 2019.
Problem 2.3.2.35 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n y^m+ b x e^{\lambda y}) w_x + y^k w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*y^m + b*x*Exp[lambda*y])*D[w[x, y], x] + y^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n*y^m+ b *x*exp(lambda*y))*diff(w(x,y),x)+ y^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (x x^{-n} {\mathrm e}^{\frac {b n y^{-k} {\mathrm e}^{\lambda y}}{\lambda }} {\mathrm e}^{\frac {b y^{-k} \left (-\lambda y \right )^{k} \Gamma \left (-k +1\right )}{\lambda }} {\mathrm e}^{\frac {b k y^{-k} \left (-\lambda y \right )^{k} \Gamma \left (-k , -\lambda y \right )}{\lambda }} {\mathrm e}^{-\frac {b y^{-k} {\mathrm e}^{\lambda y}}{\lambda }} {\mathrm e}^{-\frac {b n y^{-k} \left (-\lambda y \right )^{k} \Gamma \left (-k +1\right )}{\lambda }} {\mathrm e}^{-\frac {b k n y^{-k} \left (-\lambda y \right )^{k} \Gamma \left (-k , -\lambda y \right )}{\lambda }}+a n \left (\int y^{-k +m} {\mathrm e}^{-\frac {\left (n -1\right ) \left (k \left (-\lambda y \right )^{k} \Gamma \left (-k , -\lambda y \right )+\left (-\lambda y \right )^{k} \Gamma \left (-k +1\right )-{\mathrm e}^{\lambda y}\right ) b y^{-k}}{\lambda }}d y \right )-a \left (\int y^{-k +m} {\mathrm e}^{-\frac {\left (n -1\right ) \left (k \left (-\lambda y \right )^{k} \Gamma \left (-k , -\lambda y \right )+\left (-\lambda y \right )^{k} \Gamma \left (-k +1\right )-{\mathrm e}^{\lambda y}\right ) b y^{-k}}{\lambda }}d y \right )\right )\]
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Added January 10, 2019.
Problem 2.3.2.36 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n y^m+ b x y^k) w_x + e^{\lambda y} w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*y^m + b*x*y^k)*D[w[x, y], x] + Exp[lambda*y]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n*y^m+ b *x*y^k)*diff(w(x,y),x)+ exp(lambda*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (x^{\frac {1}{k +1}} x^{\frac {k}{k +1}} x^{-\frac {n}{k +1}} x^{-\frac {k n}{k +1}} {\mathrm e}^{\frac {b n y^{k} \left (\lambda y \right )^{-\frac {k}{2}} \WhittakerM \left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, \lambda y \right ) {\mathrm e}^{-\frac {\lambda y}{2}}}{\left (k +1\right ) \lambda }} {\mathrm e}^{-\frac {b y^{k} \left (\lambda y \right )^{-\frac {k}{2}} \WhittakerM \left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, \lambda y \right ) {\mathrm e}^{-\frac {\lambda y}{2}}}{\left (k +1\right ) \lambda }}+a n \left (\int y^{m} {\mathrm e}^{\frac {\left (n -1\right ) b y^{k} \left (\lambda y \right )^{-\frac {k}{2}} \WhittakerM \left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, \lambda y \right ) {\mathrm e}^{-\frac {\lambda y}{2}}-\left (k +1\right ) \lambda ^{2} y}{\left (k +1\right ) \lambda }}d y \right )-a \left (\int y^{m} {\mathrm e}^{\frac {\left (n -1\right ) b y^{k} \left (\lambda y \right )^{-\frac {k}{2}} \WhittakerM \left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, \lambda y \right ) {\mathrm e}^{-\frac {\lambda y}{2}}-\left (k +1\right ) \lambda ^{2} y}{\left (k +1\right ) \lambda }}d y \right )\right )\]
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