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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\text {ExpIntegralEi}\left (\frac {2 a e^{\lambda x}}{\lambda }\right ) \left (y-a e^{\lambda x}\right )+\lambda e^{\frac {2 a e^{\lambda x}}{\lambda }}}{a e^{\lambda x}-y}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {(-a{{\rm e}^{\lambda \,x}}+y) \left ( \left ( a{{\rm e}^{\lambda \,x}}-y \right ) \Ei \left ( 1,-2\,{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) +{{\rm e}^{2\,{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\lambda \right ) ^{-1}} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2^{b/\lambda } \lambda ^{-\frac {b}{\lambda }} e^{b x} a^{b/\lambda } \left (\left (a \left (-e^{\lambda x}\right )+b+y\right ) \text {LaguerreL}\left (-\frac {b}{\lambda },\frac {b}{\lambda },\frac {2 a e^{\lambda x}}{\lambda }\right )-2 a e^{\lambda x} \text {LaguerreL}\left (-\frac {b+\lambda }{\lambda },\frac {b+\lambda }{\lambda },\frac {2 a e^{\lambda x}}{\lambda }\right )\right )}{a e^{\lambda x}-y}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a{{\rm e}^{\lambda \,x}}-y} \left ( \left ( -a{{\rm e}^{\lambda \,x}}+y \right ) \int \!{{\rm e}^{{\frac {bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x+{{\rm e}^{{\frac {bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}} \right ) } \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (2 b (-1)^{-\frac {b}{\lambda }} \left (-\frac {\text {Gamma}\left (\frac {2 b}{\lambda },0,-\frac {a e^{\lambda x}}{\lambda }\right )}{\lambda }+\frac {\lambda ^{-\frac {2 b}{\lambda }} a^{\frac {2 b}{\lambda }} e^{\frac {a e^{\lambda x}+2 b \lambda x+2 i \pi b}{\lambda }}}{b-y}\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b-y} \left ( \left ( b-y \right ) \int \!{{\rm e}^{{\frac {2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x-{{\rm e}^{{\frac {2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}} \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{\lambda }^{2}x \left ( xy-1 \right ) } \left ( \left ( y{x}^{2}-x \right ) \int \!{\frac {1}{{x}^{2}}{{\rm e}^{{\frac {{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}}\,{\rm d}x-{{\rm e}^{{\frac {{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}} \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {\lambda }{\sqrt {4\,{\lambda }^{2}ab-{\lambda }^{4}}} \left ( 2\,\lambda \,\arctan \left ( {\frac {2\,\lambda \,{{\rm e}^{\lambda \,x}}ay+{\lambda }^{2}}{\sqrt {4\,{\lambda }^{2}ab-{\lambda }^{4}}}} \right ) -\sqrt {4\,{\lambda }^{2}ab-{\lambda }^{4}}x \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{\frac {b+\lambda }{\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }} \left ( \left ( b+\lambda \right ) \arctan \left ( {\frac { \left ( b+\lambda \right ) \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b+\lambda \right ) }{\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }}} \right ) -1/2\,x\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) } \right ) } \right ) \]
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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 kernel error generated
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \left ( -\sinh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{\lambda \,x}}y-\cosh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{x \left ( \lambda +\mu \right ) }}b \right ) \left ( \cosh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{\lambda \,x}}y+\sinh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{x \left ( \lambda +\mu \right ) }}b \right ) ^{-1} \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left ((-1)^{\lambda /\mu } \mu ^{-\frac {\lambda }{\mu }} a^{\lambda /\mu } \text {Gamma}\left (-\frac {\lambda }{\mu },-\frac {a e^{\mu x}}{\mu }\right )-\frac {\mu e^{\frac {a e^{\mu x}}{\mu }-\lambda x}}{y e^{\lambda x}+\lambda }\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {{{\rm e}^{\lambda \,x}} \left ( -\lambda +\mu \right ) \left ( y{{\rm e}^{\lambda \,x}}+\lambda \right ) \left ( \hypergeom \left ( [{\frac {-\lambda +\mu }{\mu }}],[{\frac {2\,\mu -\lambda }{\mu }}],{\frac {a{{\rm e}^{\mu \,x}}}{\mu }} \right ) {{\rm e}^{\mu \,x}}a\lambda -\hypergeom \left ( [-{\frac {\lambda }{\mu }}],[{\frac {-\lambda +\mu }{\mu }}],{\frac {a{{\rm e}^{\mu \,x}}}{\mu }} \right ) y{{\rm e}^{\lambda \,x}} \left ( -\lambda +\mu \right ) \right ) ^{-1}} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\mu \left (a e^{\mu x} \text {LaguerreL}\left (-\frac {-\lambda ^2+\lambda +\mu }{\mu },\frac {\lambda +\mu }{\mu },\frac {a e^{\mu x}}{\mu }\right )+\lambda \left (y e^{\lambda x}-1\right ) \text {LaguerreL}\left (\frac {(\lambda -1) \lambda }{\mu },\frac {\lambda }{\mu },\frac {a e^{\mu x}}{\mu }\right )\right )}{\lambda \left (a (\lambda -1) e^{\mu x} \text {HypergeometricU}\left (\frac {-\lambda ^2+\lambda +\mu }{\mu },\frac {\lambda }{\mu }+2,\frac {a e^{\mu x}}{\mu }\right )+\left (\mu -\mu y e^{\lambda x}\right ) \text {HypergeometricU}\left (-\frac {(\lambda -1) \lambda }{\mu },\frac {\lambda +\mu }{\mu },\frac {a e^{\mu x}}{\mu }\right )\right )}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \left ( \left ( -{\lambda }^{2}-\mu \right ) {{\sl M}\left ({\frac {-{\lambda }^{2}+\lambda -\mu }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}-{{\sl M}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )} \left ( -{{\rm e}^{\lambda \,x}}y\lambda +a{{\rm e}^{\mu \,x}}-{\lambda }^{2}-\mu +\lambda \right ) \right ) \left ( -{{\sl U}\left ({\frac {-{\lambda }^{2}+\lambda -\mu }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu +{{\sl U}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )} \left ( -{{\rm e}^{\lambda \,x}}y\lambda +a{{\rm e}^{\mu \,x}}-{\lambda }^{2}-\mu +\lambda \right ) \right ) ^{-1} \right ) \]
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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 ) ={\it \_F1} \left ( -{a \left ( y{{\rm e}^{\lambda \,x}}+{{\rm e}^{x \left ( \lambda +\mu \right ) }}b \right ) {{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }}}} \left ( 2\, \left ( 3/2\,\lambda +\mu \right ) {{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-3\, \left ( 2/3\,\lambda +\mu \right ) \left ( \lambda +\mu \right ) x}{2\,\lambda +2\,\mu }}}} \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 {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) +2\, \left ( -1/2\, \left ( 2\,\lambda +\mu \right ) \left ( \lambda +\mu \right ) {{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-3\, \left ( 2/3\,\lambda +\mu \right ) \left ( \lambda +\mu \right ) x}{2\,\lambda +2\,\mu }}}}+a \left ( b \left ( 3/2\,\lambda +\mu \right ) {{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-x\mu \, \left ( \lambda +\mu \right ) }{2\,\lambda +2\,\mu }}}}+1/2\,{{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-3\,x\mu \, \left ( \lambda +\mu \right ) }{2\,\lambda +2\,\mu }}}}y \left ( 2\,\lambda +\mu \right ) \right ) \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 {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) + \left ( \lambda +\mu \right ) \left ( - \left ( 2\,\lambda +\mu \right ) \left ( \lambda +\mu \right ) {{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-3\, \left ( 2/3\,\lambda +\mu \right ) \left ( \lambda +\mu \right ) x}{2\,\lambda +2\,\mu }}}}+ \left ( a{{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-x \left ( \lambda +\mu \right ) \left ( -2\,\lambda +\mu \right ) }{2\,\lambda +2\,\mu }}}}by+a{{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}+x \left ( 2\,\lambda +\mu \right ) \left ( \lambda +\mu \right ) }{2\,\lambda +2\,\mu }}}}{b}^{2}+ \left ( 2\,\lambda +\mu \right ) \left ( b{{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-x\mu \, \left ( \lambda +\mu \right ) }{2\,\lambda +2\,\mu }}}}+{{\rm e}^{{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-3\,x\mu \, \left ( \lambda +\mu \right ) }{2\,\lambda +2\,\mu }}}}y \right ) \right ) a \right ) \WhittakerM \left ( -{\frac {\mu }{2\,\lambda +2\,\mu }},{\frac {3\,\lambda +2\,\mu }{2\,\lambda +2\,\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \right ) ^{-1}} \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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {b}{\sqrt {4\,ac{b}^{2}-{b}^{4}} \left ( \lambda +\mu \right ) } \left ( -2\,b \left ( \lambda +\mu \right ) \arctan \left ( {\frac {2\,{{\rm e}^{\lambda \,x}}ayb+{b}^{2}}{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) +\sqrt {4\,ac{b}^{2}-{b}^{4}}{{\rm e}^{x \left ( \lambda +\mu \right ) }} \right ) } \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{{\rm e}^{x \left ( \lambda +\mu \right ) }}b+y{{\rm e}^{\lambda \,x}}+\lambda }{\lambda \, \left ( y-b{{\rm e}^{\mu \,x}} \right ) }} \right ) \]
____________________________________________________________________________________
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=()
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{y{{\rm e}^{\lambda \,x}}+\lambda } \left ( \left ( -y{{\rm e}^{\lambda \,x}}-\lambda \right ) \int \!{{\rm e}^{{\frac { \left ( {x}^{n}a-\lambda \, \left ( n+1 \right ) \right ) x}{n+1}}}}\,{\rm d}x-{{\rm e}^{{\frac { \left ( {x}^{n}a-\lambda \, \left ( n+1 \right ) \right ) x}{n+1}}}} \right ) } \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{a \left ( b{x}^{n}-y \right ) \left ( a \left ( b{x}^{n}-y \right ) \int \!\lambda \,{{\rm e}^{{\frac {abn{x}^{n} \left ( -\lambda \right ) ^{n} \left ( -\lambda \right ) ^{-n} \left ( \Gamma \left ( n,-\lambda \,x \right ) -\Gamma \left ( n \right ) \right ) \left ( -\lambda \,x \right ) ^{-n}+a{{\rm e}^{\lambda \,x}}b{x}^{n} \left ( -\lambda \right ) ^{n} \left ( -\lambda \right ) ^{-n}+{\lambda }^{2}x}{\lambda }}}}\,{\rm d}x-\lambda \,{{\rm e}^{{\frac { \left ( n \left ( \Gamma \left ( n,-\lambda \,x \right ) -\Gamma \left ( n \right ) \right ) \left ( -\lambda \,x \right ) ^{-n}+{{\rm e}^{\lambda \,x}} \right ) \left ( -\lambda \right ) ^{n}b \left ( -\lambda \right ) ^{-n}a{x}^{n}}{\lambda }}}} \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left (-i \left (a b (-1)^{-n} \lambda ^{-n-1} \text {Gamma}(n+1,-\lambda x)+\tanh ^{-1}\left (\frac {y e^{-\lambda x}}{b}\right )\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\lambda } \left ( -i\lambda \,\arctanh \left ( {\frac {{{\rm e}^{-\lambda \,x}}y}{b}} \right ) -ib \left ( \left ( \Gamma \left ( n,-\lambda \,x \right ) n-\Gamma \left ( n+1 \right ) \right ) \left ( -\lambda \,x \right ) ^{-n}+{{\rm e}^{\lambda \,x}} \right ) a{x}^{n} \right ) } \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left (\sqrt {a} \sqrt {c} \left ((-1)^{1-n} \lambda ^{-n-1} \text {Gamma}(n+1,-\lambda x)-\frac {2 \tan ^{-1}\left (\frac {b-2 a y e^{\lambda x}}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{\frac {b}{\sqrt {4\,ac{b}^{2}-{b}^{4}}\lambda } \left ( -1/2\,{x}^{n} \left ( \left ( n\Gamma \left ( n,-\lambda \,x \right ) -\Gamma \left ( n+1 \right ) \right ) \left ( -\lambda \,x \right ) ^{-n}+{{\rm e}^{\lambda \,x}} \right ) \sqrt {4\,ac{b}^{2}-{b}^{4}}+b\lambda \,\arctan \left ( {\frac {2\,{{\rm e}^{\lambda \,x}}ayb+{b}^{2}}{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) \right ) } \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {a{{\rm e}^{\lambda \,x}} \left ( b{x}^{n}+c-y \right ) -\lambda }{\lambda \, \left ( b{x}^{n}+c-y \right ) }} \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {1}{\sqrt {\lambda }} \left ( ab\sqrt {\pi }\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {\lambda }x \right ) -2\,\arctan \left ( {\frac {{{\rm e}^{-1/2\,\lambda \,{x}^{2}}}y}{b}} \right ) \sqrt {\lambda } \right ) } \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y e^{-\frac {\lambda x^2}{2}}}{b}\right )-i a b i^{-n} 2^{\frac {n-1}{2}} \lambda ^{-\frac {n}{2}-\frac {1}{2}} \text {Gamma}\left (\frac {n+1}{2},-\frac {\lambda x^2}{2}\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {2}^{n/2-1/2}{x}^{n+1}ab \left ( -\lambda \,{x}^{2} \right ) ^{-n/2-1/2}\Gamma \left ( n/2+1/2 \right ) -{2}^{n/2-1/2}{x}^{n+1}ab \left ( -\lambda \,{x}^{2} \right ) ^{-n/2-1/2}\Gamma \left ( n/2+1/2,-1/2\,\lambda \,{x}^{2} \right ) -\arctan \left ( {\frac {{{\rm e}^{-1/2\,\lambda \,{x}^{2}}}y}{b}} \right ) \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( x-\sum _{{\it \_R}=\RootOf \left ( a{{\it \_Z}}^{3}+b{{\it \_Z}}^{2}+ \left ( c+\lambda \right ) {\it \_Z}+d \right ) }{\frac {\ln \left ( y{{\rm e}^{\lambda \,x}}-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}a+2\,{\it \_R}\,b+c+\lambda }} \right ) \] 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( b+y \right ) ^{2}} \left ( 2\,a \left ( b+y \right ) ^{2}\int \!{{\rm e}^{{\frac {-6\,a{{\rm e}^{\lambda \,x}}{b}^{2}+2\,x\lambda \, \left ( c+\lambda /2 \right ) }{\lambda }}}}\,{\rm d}x+{{\rm e}^{2\,cx-6\,{\frac {a{{\rm e}^{\lambda \,x}}{b}^{2}}{\lambda }}}} \right ) } \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left (a \sqrt {b^2} x^k (-\lambda x)^{-k} \text {Gamma}(k,-\lambda x)+\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \Gamma \left ( k \right ) {x}^{k}ab \left ( -\lambda \,x \right ) ^{-k}-\Gamma \left ( k,-\lambda \,x \right ) {x}^{k}ab \left ( -\lambda \,x \right ) ^{-k}-\arctan \left ( {\frac {y{x}^{-k}}{b}} \right ) \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left (c (-\lambda x)^{-n} \sqrt {\frac {a x^{2 n}}{c}} \text {Gamma}(n,-\lambda x)-\frac {2 \sqrt {a} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \left (\sqrt {\frac {b^2}{a c}}-2 y \sqrt {\frac {a x^{2 n}}{c}}\right )}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{\frac {b}{\sqrt {4\,ac{b}^{2}-{b}^{4}}} \left ( 1/2\, \left ( -\lambda \,x \right ) ^{-n}\sqrt {4\,ac{b}^{2}-{b}^{4}}{x}^{n} \left ( \Gamma \left ( n,-\lambda \,x \right ) -\Gamma \left ( n \right ) \right ) +b\arctan \left ( {\frac {2\,bay{x}^{n}+{b}^{2}}{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {1}{a} \left ( -\sqrt {{\frac {-{b}^{2}-4\,\lambda \,c}{{a}^{2}}}}\int ^{2\,\arctan \left ( 2\,{\frac {a\lambda \,x-y\lambda \,{{\rm e}^{-\lambda \,x}}+b/2}{a}{\frac {1}{\sqrt {{\frac {-{b}^{2}-4\,\lambda \,c}{{a}^{2}}}}}}} \right ) {\frac {1}{\sqrt {{\frac {-{b}^{2}-4\,\lambda \,c}{{a}^{2}}}}}}}\!\tan \left ( 1/2\,{\it \_a}\,\sqrt {{\frac {-{b}^{2}-4\,\lambda \,c}{{a}^{2}}}} \right ) {{\rm e}^{-{\it \_a}}}{d{\it \_a}}a-{{\rm e}^{-2\,\arctan \left ( 2\,{\frac {a\lambda \,x-y\lambda \,{{\rm e}^{-\lambda \,x}}+b/2}{a}{\frac {1}{\sqrt {{\frac {-{b}^{2}-4\,\lambda \,c}{{a}^{2}}}}}}} \right ) {\frac {1}{\sqrt {{\frac {-{b}^{2}-4\,\lambda \,c}{{a}^{2}}}}}}}} \left ( 2\,a\lambda \,x+b \right ) \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{y}^{-m+1}\lambda \,a-b{{\rm e}^{-\lambda \,x}} \left ( m-1 \right ) }{a\lambda }} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{{\rm e}^{y \left ( -2\,b+1 \right ) }} \left ( x \left ( b-1 \right ) {{\rm e}^{y \left ( b-1 \right ) }}+a{{\rm e}^{by}} \right ) }{b-1}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {{x}^{ \left ( m+1 \right ) ^{-1}}{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}}{x}^{{\frac {m}{m+1}}} \left ( {x}^{{\frac {mn}{m+1}}} \right ) ^{-1} \left ( {x}^{{\frac {n}{m+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac {b{y}^{m} \left ( \mu \,y \right ) ^{-m/2}{{\rm e}^{-1/2\,\mu \,y}} \left ( n-1 \right ) \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -\mu \,y \left ( -\lambda +\mu \right ) \left ( m+1 \right ) }{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y-a\int \!{{\rm e}^{{\frac {b{y}^{m} \left ( \mu \,y \right ) ^{-m/2}{{\rm e}^{-1/2\,\mu \,y}} \left ( n-1 \right ) \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -\mu \,y \left ( -\lambda +\mu \right ) \left ( m+1 \right ) }{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {x}{{x}^{n}}{{\rm e}^{{\frac {{{\rm e}^{y\lambda }}bn{y}^{-k}}{\lambda }}}}{{\rm e}^{{\frac { \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) bk{y}^{-k}}{\lambda }}}}{{\rm e}^{{\frac {{y}^{-k} \left ( -y\lambda \right ) ^{k}b\Gamma \left ( 1-k \right ) }{\lambda }}}} \left ( {{\rm e}^{{\frac { \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) bkn{y}^{-k}}{\lambda }}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{y}^{-k} \left ( -y\lambda \right ) ^{k}b\Gamma \left ( 1-k \right ) n}{\lambda }}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{{\rm e}^{y\lambda }}b{y}^{-k}}{\lambda }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{-{\frac { \left ( n-1 \right ) \left ( k \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) + \left ( -y\lambda \right ) ^{k}\Gamma \left ( 1-k \right ) -{{\rm e}^{y\lambda }} \right ) {y}^{-k}b}{\lambda }}}}{y}^{-k+m}\,{\rm d}y-a\int \!{{\rm e}^{-{\frac { \left ( n-1 \right ) \left ( k \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) + \left ( -y\lambda \right ) ^{k}\Gamma \left ( 1-k \right ) -{{\rm e}^{y\lambda }} \right ) {y}^{-k}b}{\lambda }}}}{y}^{-k+m}\,{\rm d}y \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {{x}^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}}{x}^{{\frac {k}{k+1}}} \left ( {x}^{{\frac {kn}{k+1}}} \right ) ^{-1} \left ( {x}^{{\frac {n}{k+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac {b{y}^{k} \left ( y\lambda \right ) ^{-k/2}{{\rm e}^{-1/2\,y\lambda }} \left ( n-1 \right ) \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -y{\lambda }^{2} \left ( k+1 \right ) }{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y-a\int \!{{\rm e}^{{\frac {b{y}^{k} \left ( y\lambda \right ) ^{-k/2}{{\rm e}^{-1/2\,y\lambda }} \left ( n-1 \right ) \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -y{\lambda }^{2} \left ( k+1 \right ) }{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y \right ) \]
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