44 HFOPDE, chapter 2.3.2

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44.1 problem number 1

problem number 353

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{(a{{\rm e}^{\lambda \,x}}-y) \left ( \Ei \left ( 1,-2\,{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) {{\rm e}^{\lambda \,x}}a+{{\rm e}^{2\,{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\lambda -\Ei \left ( 1,-2\,{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) y \right ) ^{-1}} \right ) \]

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44.2 problem number 2

problem number 354

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {2^{b/\lambda } \lambda ^{-\frac {b}{\lambda }} e^{b x} a^{b/\lambda } \left (-y \text {LaguerreL}\left (-\frac {b}{\lambda },\frac {b}{\lambda },\frac {2 a e^{\lambda x}}{\lambda }\right )+2 a e^{\lambda x} \text {LaguerreL}\left (-\frac {b}{\lambda }-1,\frac {b}{\lambda }+1,\frac {2 a e^{\lambda x}}{\lambda }\right )+a e^{\lambda x} \text {LaguerreL}\left (-\frac {b}{\lambda },\frac {b}{\lambda },\frac {2 a e^{\lambda x}}{\lambda }\right )-b \text {LaguerreL}\left (-\frac {b}{\lambda },\frac {b}{\lambda },\frac {2 a e^{\lambda x}}{\lambda }\right )\right )}{a e^{\lambda x}-y}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a{{\rm e}^{\lambda \,x}}-y} \left ( {{\rm e}^{{\frac {bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}}-a{{\rm e}^{\lambda \,x}}\int \!{{\rm e}^{{\frac {bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x+y\int \!{{\rm e}^{{\frac {bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x \right ) } \right ) \]

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44.3 problem number 3

problem number 355

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{-b+y} \left ( {{\rm e}^{{\frac {2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}}+y\int \!{{\rm e}^{{\frac {2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x-b\int \!{{\rm e}^{{\frac {2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x \right ) } \right ) \]

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44.4 problem number 4

problem number 356

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{\lambda }^{2}x \left ( yx-1 \right ) } \left ( y{x}^{2}\int \!{\frac {1}{{x}^{2}}{{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}} \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}}\,{\rm d}x-{{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}} \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}-x\int \!{\frac {1}{{x}^{2}}{{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}} \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}}\,{\rm d}x \right ) } \right ) \]

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44.5 problem number 5

problem number 357

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{x \left (-\sqrt {\lambda ^2-4 a b}\right )} \left (\sqrt {\lambda ^2-4 a b}-2 a y e^{\lambda x}-\lambda \right )}{2 a y e^{\lambda x} \sqrt {\lambda ^2-4 a b}+\lambda \sqrt {\lambda ^2-4 a b}-4 a b+\lambda ^2}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {\lambda }{\sqrt {{\lambda }^{2} \left ( 4\,ab-{\lambda }^{2} \right ) }} \left ( 2\,\lambda \,\arctan \left ( {\frac {\lambda \, \left ( 2\,{{\rm e}^{\lambda \,x}}ay+\lambda \right ) }{\sqrt {4\,{\lambda }^{2}ab-{\lambda }^{4}}}} \right ) -\sqrt {{\lambda }^{2} \left ( 4\,ab-{\lambda }^{2} \right ) }x \right ) } \right ) \]

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44.6 problem number 6

problem number 358

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ \text { sol=() } \]

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44.7 problem number 7

problem number 359

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}-2 a y e^{\lambda x}-b-\lambda }{2 a y \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2} e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+\lambda x}+b^2 e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}+2 b \lambda e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}+b \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2} e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}-4 a c e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}+\lambda ^2 e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}+\lambda \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2} e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }} \left ( \sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }bx+\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }\lambda \,x-2\,{b}^{2}\arctan \left ( {\frac {2\,{{\rm e}^{\lambda \,x}}ayb+2\,\lambda \,{{\rm e}^{\lambda \,x}}ay+{b}^{2}+2\,\lambda \,b+{\lambda }^{2}}{\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }}} \right ) -4\,b\lambda \,\arctan \left ( {\frac {2\,{{\rm e}^{\lambda \,x}}ayb+2\,\lambda \,{{\rm e}^{\lambda \,x}}ay+{b}^{2}+2\,\lambda \,b+{\lambda }^{2}}{\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }}} \right ) -2\,{\lambda }^{2}\arctan \left ( {\frac {2\,{{\rm e}^{\lambda \,x}}ayb+2\,\lambda \,{{\rm e}^{\lambda \,x}}ay+{b}^{2}+2\,\lambda \,b+{\lambda }^{2}}{\sqrt { \left ( b+\lambda \right ) ^{2} \left ( 4\,ca-{b}^{2}-2\,\lambda \,b-{\lambda }^{2} \right ) }}} \right ) \right ) } \right ) \]

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44.8 problem number 8

problem number 360

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \] kernel error generated

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( {{\rm e}^{\lambda \,x}}\sinh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) y+{{\rm e}^{x \left ( \lambda +\mu \right ) }}\cosh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) b \right ) \left ( {{\rm e}^{\lambda \,x}}\cosh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) y+{{\rm e}^{x \left ( \lambda +\mu \right ) }}\sinh \left ( {\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) b \right ) ^{-1}} \right ) \]

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44.9 problem number 9

problem number 361

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{{{\rm e}^{\lambda \,x}} \left ( {{\rm e}^{\lambda \,x}}y\lambda -{{\rm e}^{\lambda \,x}}y\mu +{\lambda }^{2}-\lambda \,\mu \right ) \left ( {{\rm e}^{\lambda \,x}}y{\mbox {$_1$F$_1$}(-{\frac {\lambda }{\mu }};\,-{\frac {\lambda -\mu }{\mu }};\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }})}\lambda -{{\rm e}^{\lambda \,x}}y{\mbox {$_1$F$_1$}(-{\frac {\lambda }{\mu }};\,-{\frac {\lambda -\mu }{\mu }};\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }})}\mu +{{\rm e}^{\mu \,x}}{\mbox {$_1$F$_1$}(-{\frac {\lambda -\mu }{\mu }};\,-{\frac {-2\,\mu +\lambda }{\mu }};\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }})}a\lambda \right ) ^{-1}} \right ) \]

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44.10 problem number 10

problem number 362

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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 ( -{1 \left ( {{\rm e}^{\lambda \,x}}y{{\sl M}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda -{{\rm e}^{\mu \,x}}{{\sl M}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}a+{{\sl M}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}-{{\sl M}\left (-{\frac {{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}-{{\sl M}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda +{{\sl M}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu -{{\sl M}\left (-{\frac {{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu \right ) \left ( {{\rm e}^{\lambda \,x}}y{{\sl U}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda -{{\rm e}^{\mu \,x}}{{\sl U}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}a+{{\sl U}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}-{{\sl U}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda +{{\sl U}\left (-{\frac {\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu +{{\sl U}\left (-{\frac {{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac {\lambda +\mu }{\mu }},\,{\frac {a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu \right ) ^{-1}} \right ) \]

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44.11 problem number 11

problem number 363

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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 ( \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) y{a}^{2}b\lambda \,{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) y{a}^{2}b\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+4\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ya\lambda \,\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+3\, \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ya\lambda \,\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab{\lambda }^{2}+2\, \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\lambda }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\mu }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+4\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ya{\lambda }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ya{\mu }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\, \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ya{\lambda }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ya{\mu }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+{a}^{2}{b}^{2} \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \lambda \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+{a}^{2}{b}^{2} \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+4\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\lambda }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\mu }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}- \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\mu }^{3}+12\, \WhittakerM \left ( 1/2\,{\frac {4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{3}+2\, \WhittakerM \left ( 1/2\,{\frac {4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\mu }^{3}-2\,{{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {\lambda }^{3}-{{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {\mu }^{3}-4\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{3}-8\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{2}\mu -5\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}\lambda \,{\mu }^{2}+20\, \WhittakerM \left ( 1/2\,{\frac {4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{2}\mu +11\, \WhittakerM \left ( 1/2\,{\frac {4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}\lambda \,{\mu }^{2}-5\,{{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {\lambda }^{2}\mu -4\,{{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \lambda \,{\mu }^{2}+3\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab\lambda \,\mu +4\, \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab\lambda \,\mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+3\, \WhittakerM \left ( -1/2\,{\frac {\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab\lambda \,\mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,{\lambda }^{2}x-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( 1/2\,{\frac {2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac {3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) {{\rm e}^{1/2\,{\frac {ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab{\mu }^{2} \right ) ^{-1}} \right ) \]

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44.12 problem number 12

problem number 364

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {i \pi e^{-\frac {\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}} \left (\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}-2 a y e^{x (2 \lambda +\mu )}+b \left (-e^{x (\lambda +\mu )}\right )\right )}{2 \left (2 a y e^{x (2 \lambda +\mu )} \cosh \left (\frac {\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )+\sqrt {b^2-4 a c} e^{x (\lambda +\mu )} \sinh \left (\frac {\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )+b e^{x (\lambda +\mu )} \cosh \left (\frac {\sqrt {b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )\right )}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \left ( \lambda +\mu \right ) } \left ( 2\,b\lambda \,\arctan \left ( {\frac {b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) +2\,b\mu \,\arctan \left ( {\frac {b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) -{{\rm e}^{x \left ( \lambda +\mu \right ) }}\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \right ) } \right ) \]

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44.13 problem number 13

problem number 365

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

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

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44.14 problem number 14

problem number 366

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ \text { sol=() } \]

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44.15 problem number 15

problem number 367

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{y{{\rm e}^{\lambda \,x}}+\lambda } \left ( {{\rm e}^{\lambda \,x}}y\int \!{{\rm e}^{{\frac {x \left ( {x}^{n}a-\lambda \,n-\lambda \right ) }{n+1}}}}\,{\rm d}x+\int \!{{\rm e}^{{\frac {x \left ( {x}^{n}a-\lambda \,n-\lambda \right ) }{n+1}}}}\,{\rm d}x\lambda +{{\rm e}^{{\frac {x \left ( {x}^{n}a-\lambda \,n-\lambda \right ) }{n+1}}}} \right ) } \right ) \]

____________________________________________________________________________________

44.16 problem number 16

problem number 368

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ \text { sol=() } \]

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44.17 problem number 17

problem number 369

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {a \left ( b{x}^{n}-y \right ) \left ( -\int \!\lambda \,{{\rm e}^{-{\frac {ab \left ( -\lambda \right ) ^{-n} \left ( {x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}+\lambda \,x}}\,{\rm d}x{x}^{n}ab+\int \!\lambda \,{{\rm e}^{-{\frac {ab \left ( -\lambda \right ) ^{-n} \left ( {x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}+\lambda \,x}}\,{\rm d}xya+\lambda \,{{\rm e}^{-{\frac {ab \left ( -\lambda \right ) ^{-n} \left ( {x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}}} \right ) ^{-1}} \right ) \]

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44.18 problem number 18

problem number 370

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ \text { sol=() } \]

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44.19 problem number 19

problem number 371

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {i}{\lambda } \left ( a\Gamma \left ( n \right ) {x}^{n} \left ( -\lambda \,x \right ) ^{-n}bn-\Gamma \left ( n,-\lambda \,x \right ) a{x}^{n} \left ( -\lambda \,x \right ) ^{-n}bn-{{\rm e}^{\lambda \,x}}{x}^{n}ab-\arctanh \left ( {\frac {{{\rm e}^{-\lambda \,x}}y}{b}} \right ) \lambda \right ) } \right ) \]

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44.20 problem number 20

problem number 372

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

44.21 problem number 21

problem number 373

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

44.22 problem number 22

problem number 374

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-4 a^{3/2} c^{3/2} (-1)^{-n} \lambda ^{-n-1} \text {Gamma}(n+1,-\lambda x)+\sqrt {a} b^2 \sqrt {c} (-1)^{-n} \lambda ^{-n-1} \text {Gamma}(n+1,-\lambda x)+2 \sqrt {a} \sqrt {c} \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {2 a y e^{\lambda x} \sqrt {4 a c-b^2}-b \sqrt {4 a c-b^2}}{4 a c-b^2}\right )}{4 a c-b^2}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }\lambda } \left ( -{{\rm e}^{\lambda \,x}}\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}- \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}n+2\,b\lambda \,\arctan \left ( {\frac {b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) + \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n \right ) \sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}n \right ) } \right ) \]

____________________________________________________________________________________

44.23 problem number 23

problem number 375

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

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

____________________________________________________________________________________

44.24 problem number 24

problem number 376

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

44.25 problem number 25

problem number 377

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \left (2 \tan ^{-1}\left (\frac {y e^{-\frac {\lambda x^2}{2}}}{b}\right )-\frac {\sqrt {2 \pi } a b \text {Erf}\left (\frac {\sqrt {\lambda } x}{\sqrt {2}}\right )}{\sqrt {\lambda }}\right )\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {1}{\sqrt {\lambda }} \left ( ab\sqrt {\pi }\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {\lambda }x \right ) -2\,\arctan \left ( {\frac {{{\rm e}^{-1/2\,\lambda \,{x}^{2}}}y}{b}} \right ) \sqrt {\lambda } \right ) } \right ) \]

____________________________________________________________________________________

44.26 problem number 26

problem number 378

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {2}^{n/2-1/2}{x}^{n+1}ab \left ( -\lambda \,{x}^{2} \right ) ^{-1/2-n/2}\Gamma \left ( n/2+1/2 \right ) -{2}^{n/2-1/2}{x}^{n+1}ab \left ( -\lambda \,{x}^{2} \right ) ^{-1/2-n/2}\Gamma \left ( n/2+1/2,-1/2\,\lambda \,{x}^{2} \right ) -\arctan \left ( {\frac {{{\rm e}^{-1/2\,\lambda \,{x}^{2}}}y}{b}} \right ) \right ) \]

____________________________________________________________________________________

44.27 problem number 27

problem number 379

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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

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44.28 problem number 28

problem number 380

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {6 a b^2 e^{\lambda x}}{\lambda }} \left (2 y^2 e^{\frac {6 a b^2 e^{\lambda x}}{\lambda }} \int _1^x a \exp \left (-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]+4 b y e^{\frac {6 a b^2 e^{\lambda x}}{\lambda }} \left (\int _1^x a \exp \left (-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]\right )+2 b^2 e^{\frac {6 a b^2 e^{\lambda x}}{\lambda }} \left (\int _1^x a \exp \left (-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]\right )+e^{2 c x}\right )}{(b+y)^2}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( b+y \right ) ^{2}} \left ( 2\,a{b}^{2}\int \!{{\rm e}^{-{\frac {6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-{\lambda }^{2}x}{\lambda }}}}\,{\rm d}x+4\,yab\int \!{{\rm e}^{-{\frac {6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-{\lambda }^{2}x}{\lambda }}}}\,{\rm d}x+2\,{y}^{2}a\int \!{{\rm e}^{-{\frac {6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-{\lambda }^{2}x}{\lambda }}}}\,{\rm d}x+{{\rm e}^{2\,cx-6\,{\frac {a{{\rm e}^{\lambda \,x}}{b}^{2}}{\lambda }}}} \right ) } \right ) \]

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44.29 problem number 29

problem number 381

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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 ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( a\Gamma \left ( k \right ) {x}^{k} \left ( -\lambda \,x \right ) ^{-k}b-\Gamma \left ( k,-\lambda \,x \right ) a{x}^{k} \left ( -\lambda \,x \right ) ^{-k}b-\arctan \left ( {\frac {y{x}^{-k}}{b}} \right ) \right ) \]

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44.30 problem number 30

problem number 382

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-b^2 c (-\lambda x)^{-n} \sqrt {\frac {a x^{2 n}}{c}} \text {Gamma}(n,-\lambda x)+4 a c^2 (-\lambda x)^{-n} \sqrt {\frac {a x^{2 n}}{c}} \text {Gamma}(n,-\lambda x)-2 \sqrt {a} \sqrt {c} \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {\frac {b^2}{a c}} \sqrt {4 a c-b^2}-2 \sqrt {a} \sqrt {c} y \sqrt {4 a c-b^2} \sqrt {\frac {a x^{2 n}}{c}}}{4 a c-b^2}\right )}{4 a c-b^2}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }} \left ( -\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}+\Gamma \left ( n,-\lambda \,x \right ) \left ( -\lambda \,x \right ) ^{-n}\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}+2\,b\arctan \left ( {\frac {b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) \right ) } \right ) \]

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44.31 problem number 31

problem number 383

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {1}{a} \left ( 2\,ax\lambda \,{{\rm e}^{2\,{1\arctan \left ( {\frac {2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,ax\lambda -b}{a}{\frac {1}{\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ) {\frac {1}{\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}}+\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}\int ^{-2\,{1\arctan \left ( {\frac {2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,ax\lambda -b}{a}{\frac {1}{\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ) {\frac {1}{\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}\!\tan \left ( 1/2\,{\it \_a}\,\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}} \right ) {{\rm e}^{-{\it \_a}}}{d{\it \_a}}a+b{{\rm e}^{2\,{1\arctan \left ( {\frac {2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,ax\lambda -b}{a}{\frac {1}{\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ) {\frac {1}{\sqrt {-{\frac {{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}} \right ) } \right ) \]

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44.32 problem number 32

problem number 384

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {b{{\rm e}^{-\lambda \,x}}m-{y}^{1-m}a\lambda -b{{\rm e}^{-\lambda \,x}}}{a\lambda }} \right ) \]

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44.33 problem number 33

problem number 385

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( x{{\rm e}^{y \left ( b-1 \right ) }}b-x{{\rm e}^{y \left ( b-1 \right ) }}+a{{\rm e}^{by}} \right ) {{\rm e}^{-y \left ( 2\,b-1 \right ) }}}{b-1}} \right ) \]

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44.34 problem number 34

problem number 386

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {{x}^{ \left ( m+1 \right ) ^{-1}}{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}}{x}^{{\frac {m}{m+1}}} \left ( {x}^{{\frac {mn}{m+1}}} \right ) ^{-1} \left ( {x}^{{\frac {n}{m+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) +\lambda \,\mu \,ym-{\mu }^{2}ym+\lambda \,\mu \,y-{\mu }^{2}y}{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y-a\int \!{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) +\lambda \,\mu \,ym-{\mu }^{2}ym+\lambda \,\mu \,y-{\mu }^{2}y}{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y \right ) \]

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44.35 problem number 35

problem number 387

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {x}{{x}^{n}}{{\rm e}^{{\frac {{y}^{-k}{{\rm e}^{y\lambda }}bn}{\lambda }}}}{{\rm e}^{{\frac { \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) bk{y}^{-k}}{\lambda }}}}{{\rm e}^{{\frac {{y}^{-k} \left ( -y\lambda \right ) ^{k}b\Gamma \left ( 1-k \right ) }{\lambda }}}} \left ( {{\rm e}^{{\frac { \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) bkn{y}^{-k}}{\lambda }}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{y}^{-k} \left ( -y\lambda \right ) ^{k}b\Gamma \left ( 1-k \right ) n}{\lambda }}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{y}^{-k}{{\rm e}^{y\lambda }}b}{\lambda }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac { \left ( n-1 \right ) b{y}^{-k} \left ( k\Gamma \left ( -k \right ) \left ( -y\lambda \right ) ^{k}-k \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) +{{\rm e}^{y\lambda }} \right ) }{\lambda }}}}{y}^{-k+m}\,{\rm d}y-a\int \!{{\rm e}^{{\frac { \left ( n-1 \right ) b{y}^{-k} \left ( k\Gamma \left ( -k \right ) \left ( -y\lambda \right ) ^{k}-k \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) +{{\rm e}^{y\lambda }} \right ) }{\lambda }}}}{y}^{-k+m}\,{\rm d}y \right ) \]

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44.36 problem number 36

problem number 388

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 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]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
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'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {{x}^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}}{x}^{{\frac {k}{k+1}}} \left ( {x}^{{\frac {kn}{k+1}}} \right ) ^{-1} \left ( {x}^{{\frac {n}{k+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -{\lambda }^{2}yk-{\lambda }^{2}y}{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y-a\int \!{{\rm e}^{{\frac {bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -{\lambda }^{2}yk-{\lambda }^{2}y}{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y \right ) \]