6.2.7 3.2

6.2.7.1 [544] problem number 1

problem number 544

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\operatorname {ExpIntegralEi}\left (\frac {2 a e^{\lambda x}}{\lambda }\right ) \left (y-a e^{\lambda x}\right )+\lambda e^{\frac {2 a e^{\lambda x}}{\lambda }}}{a e^{\lambda x}-y}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-{\mathrm e}^{\lambda x} a +y}{\left ({\mathrm e}^{\lambda x} a -y \right ) \operatorname {Ei}_{1}\left (-\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right )+{\mathrm e}^{\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }} \lambda }\right )\]

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6.2.7.2 [545] problem number 2

problem number 545

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {2^{b/\lambda } \lambda ^{-\frac {b}{\lambda }} e^{b x} a^{b/\lambda } \left (L_{-\frac {b}{\lambda }}^{\frac {b}{\lambda }}\left (\frac {2 a e^{\lambda x}}{\lambda }\right ) \left (a \left (-e^{\lambda x}\right )+b+y\right )-2 a e^{\lambda x} L_{-\frac {b+\lambda }{\lambda }}^{\frac {b+\lambda }{\lambda }}\left (\frac {2 a e^{\lambda x}}{\lambda }\right )\right )}{a e^{\lambda x}-y}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-{\mathrm e}^{\lambda x} a +y \right ) \int {\mathrm e}^{\frac {b x \lambda +2 \,{\mathrm e}^{\lambda x} a}{\lambda }}d x +{\mathrm e}^{\frac {b x \lambda +2 \,{\mathrm e}^{\lambda x} a}{\lambda }}}{{\mathrm e}^{\lambda x} a -y}\right )\]

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6.2.7.3 [546] problem number 3

problem number 546

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (2 b (-1)^{-\frac {b}{\lambda }} \left (-\frac {\Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{\lambda x}}{\lambda }\right )}{\lambda }+\frac {\lambda ^{-\frac {2 b}{\lambda }} a^{\frac {2 b}{\lambda }} e^{\frac {a e^{\lambda x}+2 b \lambda x+2 i \pi b}{\lambda }}}{b-y}\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (b -y \right ) \int {\mathrm e}^{\frac {2 x b \lambda +{\mathrm e}^{\lambda x} a}{\lambda }}d x -{\mathrm e}^{\frac {2 x b \lambda +{\mathrm e}^{\lambda x} a}{\lambda }}}{b -y}\right )\]

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6.2.7.4 [547] problem number 4

problem number 547

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\int \frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x} \left (\lambda x -1\right )}{\lambda ^{2}}}}{x^{2}}d x x^{2} y -\int \frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x} \left (\lambda x -1\right )}{\lambda ^{2}}}}{x^{2}}d x x -{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x} \left (\lambda x -1\right )}{\lambda ^{2}}}}{\lambda ^{2} x \left (x y -1\right )}\right )\]

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6.2.7.5 [548] problem number 5

problem number 548

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 \lambda \left (\lambda \arctan \left (\frac {2 \,{\mathrm e}^{\lambda x} a \lambda y +\lambda ^{2}}{\sqrt {4 \lambda ^{2} a b -\lambda ^{4}}}\right )-\frac {\sqrt {4 \lambda ^{2} a b -\lambda ^{4}}\, x}{2}\right )}{\sqrt {4 \lambda ^{2} a b -\lambda ^{4}}}\right )\]

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6.2.7.6 [549] problem number 6

problem number 549

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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6.2.7.7 [550] problem number 7

problem number 550

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 \left (b +\lambda \right ) \left (\left (b +\lambda \right ) \arctan \left (\frac {\left (b +\lambda \right ) \left (2 \,{\mathrm e}^{\lambda x} y a +b +\lambda \right )}{\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}}\right )-\frac {x \sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}}{2}\right )}{\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}}\right )\]

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6.2.7.8 [551] problem number 8

problem number 551

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-{\mathrm e}^{x \mu } \cosh \left (\frac {a b \,{\mathrm e}^{x \left (\mu +\lambda \right )}}{\mu +\lambda }\right ) b -y \sinh \left (\frac {a b \,{\mathrm e}^{x \left (\mu +\lambda \right )}}{\mu +\lambda }\right )}{y \cosh \left (\frac {a b \,{\mathrm e}^{x \left (\mu +\lambda \right )}}{\mu +\lambda }\right )+{\mathrm e}^{x \mu } \sinh \left (\frac {a b \,{\mathrm e}^{x \left (\mu +\lambda \right )}}{\mu +\lambda }\right ) b}\right )\]

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6.2.7.9 [552] problem number 9

problem number 552

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left ((-1)^{\lambda /\mu } \mu ^{-\frac {\lambda }{\mu }} a^{\lambda /\mu } \Gamma \left (-\frac {\lambda }{\mu },-\frac {a e^{\mu x}}{\mu }\right )-\frac {\mu e^{\frac {a e^{\mu x}}{\mu }-\lambda x}}{y e^{\lambda x}+\lambda }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {{\mathrm e}^{\lambda x} \left (\mu -\lambda \right ) \left ({\mathrm e}^{\lambda x} y +\lambda \right )}{{\mathrm e}^{\mu x} \operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) a \lambda -y \,{\mathrm e}^{\lambda x} \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) \left (\mu -\lambda \right )}\right )\]

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6.2.7.10 [553] problem number 10

problem number 553

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\mu \left (a e^{\mu x} L_{-\frac {-\lambda ^2+\lambda +\mu }{\mu }}^{\frac {\lambda +\mu }{\mu }}\left (\frac {a e^{\mu x}}{\mu }\right )+\lambda \left (y e^{\lambda x}-1\right ) L_{\frac {(\lambda -1) \lambda }{\mu }}^{\frac {\lambda }{\mu }}\left (\frac {a e^{\mu x}}{\mu }\right )\right )}{\lambda \left (a (\lambda -1) e^{\mu x} \operatorname {HypergeometricU}\left (\frac {-\lambda ^2+\lambda +\mu }{\mu },\frac {\lambda }{\mu }+2,\frac {a e^{\mu x}}{\mu }\right )+\left (\mu -\mu y e^{\lambda x}\right ) \operatorname {HypergeometricU}\left (-\frac {(\lambda -1) \lambda }{\mu },\frac {\lambda +\mu }{\mu },\frac {a e^{\mu x}}{\mu }\right )\right )}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\mu \left (\left (\lambda -1\right ) \operatorname {KummerM}\left (\frac {-\lambda ^{2}+\lambda +\mu }{\mu }, \frac {\mu +\lambda }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+\operatorname {KummerM}\left (-\frac {\lambda \left (\lambda -1\right )}{\mu }, \frac {\mu +\lambda }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) \left (y \,{\mathrm e}^{\lambda x}-\lambda \right )\right )}{\lambda ^{2} \left (\lambda -1\right ) \operatorname {KummerU}\left (\frac {-\lambda ^{2}+\lambda +\mu }{\mu }, \frac {\mu +\lambda }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )-\mu \operatorname {KummerU}\left (-\frac {\lambda \left (\lambda -1\right )}{\mu }, \frac {\mu +\lambda }{\mu }, \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) \left (y \,{\mathrm e}^{\lambda x}-\lambda \right )}\right )\]

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6.2.7.11 [554] problem number 11

problem number 554

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 ) = f_{1} \left (-\frac {a \left (y +b \,{\mathrm e}^{\mu x}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+2 x \left (\lambda +\mu \right )^{2}}{\lambda +\mu }}}{\left (2 \lambda +\mu \right ) \left (2 \left (\frac {3 \lambda }{2}+\mu \right ) a b \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+3 x \left (\mu +\frac {2 \lambda }{3}\right ) \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+\left (2 \lambda +\mu \right ) \left (a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+x \left (2 \lambda +\mu \right ) \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} y -{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+\mu x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\lambda +\mu \right )\right )\right ) \operatorname {WhittakerM}\left (\frac {2 \lambda +\mu }{2 \lambda +2 \mu }, \frac {3 \lambda +2 \mu }{2 \lambda +2 \mu }, \frac {b a \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right )+\left (a b \left (2 \lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+3 x \left (\mu +\frac {2 \lambda }{3}\right ) \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+a^{2} {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+3 x \left (\mu +\frac {4 \lambda }{3}\right ) \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b y +a^{2} {\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+5 \left (\mu +\frac {4 \lambda }{5}\right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b^{2}+\left (2 \lambda +\mu \right ) \left (a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+x \left (2 \lambda +\mu \right ) \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} y -{\mathrm e}^{\frac {a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+\mu x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\lambda +\mu \right )\right )\right ) \left (\lambda +\mu \right ) \operatorname {WhittakerM}\left (-\frac {\mu }{2 \lambda +2 \mu }, \frac {3 \lambda +2 \mu }{2 \lambda +2 \mu }, \frac {b a \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right )+2 \,{\mathrm e}^{\frac {\mu x}{2}} \left (\frac {3 \lambda }{2}+\mu \right ) \left (2 \lambda +\mu \right )^{2} \left (\frac {b a \,{\mathrm e}^{\left (\lambda +\mu \right ) x}}{\lambda +\mu }\right )^{\frac {4 \lambda +3 \mu }{2 \lambda +2 \mu }}}\right )\]

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6.2.7.12 [555] problem number 12

problem number 555

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (-2 b \left (\lambda +\mu \right ) \arctan \left (\frac {b^{2}+2 \,{\mathrm e}^{x \lambda } a y b}{\sqrt {4 b^{2} a c -b^{4}}}\right )+\sqrt {4 b^{2} a c -b^{4}}\, {\mathrm e}^{\left (\lambda +\mu \right ) x}\right ) b}{\sqrt {4 b^{2} a c -b^{4}}\, \left (\lambda +\mu \right )}\right )\]

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6.2.7.13 [556] problem number 13

problem number 556

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-y +b \,{\mathrm e}^{\mu x}}{\lambda \left ({\mathrm e}^{\lambda x} y -{\mathrm e}^{x \left (\lambda +\mu \right )} b +\lambda \right )}\right )\]

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6.2.7.14 [557] problem number 14

problem number 557

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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6.2.7.15 [558] problem number 15

problem number 558

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-{\mathrm e}^{\lambda x} y -\lambda \right ) \int {\mathrm e}^{\frac {x \left (a \,x^{n}-\lambda \left (n +1\right )\right )}{n +1}}d x -{\mathrm e}^{\frac {x \left (a \,x^{n}-\lambda \left (n +1\right )\right )}{n +1}}}{{\mathrm e}^{\lambda x} y +\lambda }\right )\]

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6.2.7.16 [559] problem number 16

problem number 559

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=()

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6.2.7.17 [560] problem number 17

problem number 560

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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {a \left (x^{n} b -y \right )}{\left (a \left (x^{n} b -y \right ) \int {\mathrm e}^{\frac {\left (\left (a b \,x^{n} {\mathrm e}^{\lambda x}+\lambda ^{2} x \right ) \left (-\lambda x \right )^{n}+x^{n} a b \left (\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (n +1\right )\right )\right ) \left (-\lambda x \right )^{-n}}{\lambda }}d x -{\mathrm e}^{\frac {x^{n} a b \left (\left (-\lambda x \right )^{-n} \Gamma \left (n , -\lambda x \right ) n -\Gamma \left (n +1\right ) \left (-\lambda x \right )^{-n}+{\mathrm e}^{\lambda x}\right )}{\lambda }}\right ) \lambda }\right )\]

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6.2.7.18 [561] problem number 18

problem number 561

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=()

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6.2.7.19 [562] problem number 19

problem number 562

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-i \left (a b (-1)^{-n} \lambda ^{-n-1} \Gamma (n+1,-\lambda x)+\text {arctanh}\left (\frac {y e^{-\lambda x}}{b}\right )\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {i \left (\operatorname {arctanh}\left (\frac {{\mathrm e}^{-\lambda x} y}{b}\right ) \lambda +b a \,x^{n} \left (\left (\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (n +1\right )\right ) \left (-\lambda x \right )^{-n}+{\mathrm e}^{\lambda x}\right )\right )}{\lambda }\right )\]

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6.2.7.20 [563] problem number 20

problem number 563

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\lambda x} \left (e^{\lambda x} \left (y-b e^{\lambda x}\right ) \int _1^{e^{\lambda x}}\frac {\exp \left ((-1)^{-n} a b \lambda ^{-n-1} \Gamma (n+1,-\log (K[1]))\right )}{K[1]^2}dK[1]+y e^{a b (-1)^{-n} \lambda ^{-n-1} \Gamma (n+1,-\lambda x)}\right )}{b e^{\lambda x}-y}\right )\right \}\right \}\]

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=()

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6.2.7.21 [564] problem number 21

problem number 564

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=()

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6.2.7.22 [565] problem number 22

problem number 565

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]];
 

Failed

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (\frac {2 \arctan \left (\frac {b^{2}+2 a b \,{\mathrm e}^{\lambda x} y}{\sqrt {4 b^{2} a c -b^{4}}}\right ) b \lambda }{\sqrt {4 b^{2} a c -b^{4}}}-\left (-\lambda x \right )^{-n} \Gamma \left (n , -\lambda x \right ) x^{n} n +\left (-\lambda x \right )^{-n} x^{n} \Gamma \left (n +1\right )-x^{n} {\mathrm e}^{\lambda x}\right ) b}{\lambda }\right )\]

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6.2.7.23 [566] problem number 23

problem number 566

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {a \left (b \,x^{n}+c -y \right )}{\left ({\mathrm e}^{\lambda x} a \left (b \,x^{n}+c -y \right )-\lambda \right ) \lambda }\right )\]

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6.2.7.24 [567] problem number 24

problem number 567

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=()

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6.2.7.25 [568] problem number 25

problem number 568

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y e^{-\frac {\lambda x^2}{2}}}{b}\right )-\frac {\sqrt {\frac {\pi }{2}} a b \text {erf}\left (\frac {\sqrt {\lambda } x}{\sqrt {2}}\right )}{\sqrt {\lambda }}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {b a \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\lambda }\, x}{2}\right )-2 \arctan \left (\frac {{\mathrm e}^{-\frac {\lambda \,x^{2}}{2}} y}{b}\right ) \sqrt {\lambda }}{2 \sqrt {\lambda }}\right )\]

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6.2.7.26 [569] problem number 26

problem number 569

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y e^{-\frac {\lambda x^2}{2}}}{b}\right )-i a b i^{-n} 2^{\frac {n-1}{2}} \lambda ^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n+1}{2},-\frac {\lambda x^2}{2}\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (a b 2^{-\frac {1}{2}+\frac {n}{2}} x^{n +1} \Gamma \left (\frac {n}{2}+\frac {1}{2}\right ) \left (-\lambda \,x^{2}\right )^{-\frac {1}{2}-\frac {n}{2}}-a b 2^{-\frac {1}{2}+\frac {n}{2}} x^{n +1} \left (-\lambda \,x^{2}\right )^{-\frac {1}{2}-\frac {n}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}, -\frac {\lambda \,x^{2}}{2}\right )-\arctan \left (\frac {{\mathrm e}^{-\frac {\lambda \,x^{2}}{2}} y}{b}\right )\right )\]

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6.2.7.27 [570] problem number 27

problem number 570

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'));
 

\[w \left (x , y\right ) = f_{1} \left (x -\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \,\textit {\_Z}^{2}+\left (c +\lambda \right ) \textit {\_Z} +d \right )}{\sum }\frac {\ln \left ({\mathrm e}^{\lambda x} y -\textit {\_R} \right )}{3 \textit {\_R}^{2} a +2 \textit {\_R} b +c +\lambda }\right )\right )\]

Solution contains RootOf

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6.2.7.28 [571] problem number 28

problem number 571

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 a \left (b +y \right )^{2} \int {\mathrm e}^{\frac {-6 \,{\mathrm e}^{\lambda x} a \,b^{2}+2 \lambda x \left (c +\frac {\lambda }{2}\right )}{\lambda }}d x +{\mathrm e}^{\frac {-6 \,{\mathrm e}^{\lambda x} a \,b^{2}+2 x c \lambda }{\lambda }}}{\left (b +y \right )^{2}}\right )\]

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6.2.7.29 [572] problem number 29

problem number 572

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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (a \sqrt {b^2} x^k (-\lambda x)^{-k} \Gamma (k,-\lambda x)+\arctan \left (\frac {y x^{-k}}{\sqrt {b^2}}\right )\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\arctan \left (\frac {x^{-k} y}{b}\right )+a b \,x^{k} \left (-\Gamma \left (k , -\lambda x \right )+\Gamma \left (k \right )\right ) \left (-\lambda x \right )^{-k}\right )\]

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6.2.7.30 [573] problem number 30

problem number 573

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]];
 

Failed

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\left (\frac {2 b \arctan \left (\frac {b^{2}+2 a y \,x^{n} b}{\sqrt {4 b^{2} a c -b^{4}}}\right )}{\sqrt {4 b^{2} a c -b^{4}}}+\left (-\lambda x \right )^{-n} x^{n} \left (\Gamma \left (n , -\lambda x \right )-\Gamma \left (n \right )\right )\right ) b \right )\]

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6.2.7.31 [574] problem number 31

problem number 574

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-2 \,{\mathrm e}^{-\textit {\_a}} \arctan \left (\frac {2 \lambda a x +b -2 \,{\mathrm e}^{-\lambda x} y \lambda }{a \sqrt {\frac {-b^{2}-4 c \lambda }{a^{2}}}}\right ) a \tan \left (\frac {\textit {\_a} \sqrt {\frac {-b^{2}-4 c \lambda }{a^{2}}}}{2}\right )-{\mathrm e}^{-\frac {2 \arctan \left (\frac {2 \lambda a x +b -2 \,{\mathrm e}^{-\lambda x} y \lambda }{a \sqrt {\frac {-b^{2}-4 c \lambda }{a^{2}}}}\right )}{\sqrt {\frac {-b^{2}-4 c \lambda }{a^{2}}}}} \left (2 \lambda a x +b \right )}{2 a}\right )\]

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6.2.7.32 [575] problem number 32

problem number 575

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {a \lambda \,y^{1-m}-b \,{\mathrm e}^{-\lambda x} \left (m -1\right )}{\lambda a}\right )\]

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6.2.7.33 [576] problem number 33

problem number 576

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]];
 

\[\left \{\left \{w(x,y)\to \frac {a c_1 e^{y-b y}}{b-1}+c_1 x e^{-b y}+c_2\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {{\mathrm e}^{-y \left (b -1\right )} \left (x \left (b -1\right ) {\mathrm e}^{-y}+a \right )}{b -1}\right )\]

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6.2.7.34 [577] problem number 34

problem number 577

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {x^{-n} \left (\left (\mu \textit {\_a} \right )^{-\frac {m}{2}} \left (n -1\right ) \int \left (b x \,\textit {\_a}^{m} \left (\mu y \right )^{\frac {m}{2}} {\mathrm e}^{\frac {2 \,{\mathrm e}^{-\frac {\mu \textit {\_a}}{2}} b \,\textit {\_a}^{m} \left (\mu \textit {\_a} \right )^{-\frac {m}{2}} \operatorname {WhittakerM}\left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) \left (n -1\right )-3 \mu ^{2} \left (m +1\right ) \left (y -\frac {\textit {\_a}}{3}\right )}{2 \mu \left (m +1\right )}}-\frac {b x \,\textit {\_a}^{m} \textit {\_a} \,{\mathrm e}^{\frac {{\mathrm e}^{-\frac {\mu \textit {\_a}}{2}} b \,\textit {\_a}^{m} \left (\mu \textit {\_a} \right )^{-\frac {m}{2}} \operatorname {WhittakerM}\left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) \left (n -1\right )-\mu ^{2} y \left (m +1\right )}{\mu \left (m +1\right )}} \left (\mu \textit {\_a} \right )^{\frac {m}{2}}}{y}-a \,x^{n} \left (\mu \textit {\_a} \right )^{\frac {m}{2}} {\mathrm e}^{\frac {{\mathrm e}^{-\frac {\mu \textit {\_a}}{2}} b \,\textit {\_a}^{m} \left (\mu \textit {\_a} \right )^{-\frac {m}{2}} \operatorname {WhittakerM}\left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) \left (n -1\right )-\mu \left (-\lambda \textit {\_a} +\mu y \right ) \left (m +1\right )}{\mu \left (m +1\right )}}\right )d y -{\mathrm e}^{\frac {b \operatorname {WhittakerM}\left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) \left (n -1\right ) \left (\frac {1}{\mu }\right )^{\frac {m}{2}} {\mathrm e}^{-\frac {\mu y}{2}} y^{\frac {m}{2}}}{\mu \left (m +1\right )}} x \right )}{n -1}\right )\]

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6.2.7.35 [578] problem number 35

problem number 578

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\int y^{m -k} {\mathrm e}^{-\frac {y^{-k} b \left (n -1\right ) \left (k \left (-\lambda y \right )^{k} \Gamma \left (-k , -\lambda y \right )+\left (-\lambda y \right )^{k} \Gamma \left (-k +1\right )-{\mathrm e}^{\lambda y}\right )}{\lambda }}d y a \left (n -1\right )+{\mathrm e}^{-\frac {y^{-k} b \left (n -1\right ) \left (k \left (-\lambda y \right )^{k} \Gamma \left (-k , -\lambda y \right )+\left (-\lambda y \right )^{k} \Gamma \left (-k +1\right )-{\mathrm e}^{\lambda y}\right )}{\lambda }} x^{-n +1}\right )\]

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6.2.7.36 [579] problem number 36

problem number 579

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (\left (\lambda \textit {\_a} \right )^{-\frac {k}{2}} \left (n -1\right ) \int \frac {-b x \,\textit {\_a}^{k} \left (\lambda y \right )^{\frac {k}{2}} {\mathrm e}^{\frac {2 \,{\mathrm e}^{-\frac {\lambda \textit {\_a}}{2}} b \,\textit {\_a}^{k} \left (\lambda \textit {\_a} \right )^{-\frac {k}{2}} \operatorname {WhittakerM}\left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, \lambda y \right ) \left (n -1\right )-3 \left (k +1\right ) \left (y -\frac {\textit {\_a}}{3}\right ) \lambda ^{2}}{2 \lambda \left (k +1\right )}} y +\left (\lambda \textit {\_a} \right )^{\frac {k}{2}} {\mathrm e}^{\frac {{\mathrm e}^{-\frac {\lambda \textit {\_a}}{2}} b \,\textit {\_a}^{k} \left (\lambda \textit {\_a} \right )^{-\frac {k}{2}} \operatorname {WhittakerM}\left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, \lambda y \right ) \left (n -1\right )-\lambda ^{2} y \left (k +1\right )}{\lambda \left (k +1\right )}} \textit {\_a} \left (\textit {\_a}^{k} b x +\textit {\_a}^{m} x^{n} a \right )}{y}d y +{\mathrm e}^{\frac {b \operatorname {WhittakerM}\left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, \lambda y \right ) \left (n -1\right ) \left (\frac {1}{\lambda }\right )^{\frac {k}{2}} {\mathrm e}^{-\frac {\lambda y}{2}} y^{\frac {k}{2}}}{\lambda \left (k +1\right )}} x \right ) x^{-n}}{n -1}\right )\]

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