6.2.32 9.3

6.2.32.1 [821] problem number 1
6.2.32.2 [822] problem number 2
6.2.32.3 [823] problem number 3
6.2.32.4 [824] problem number 4
6.2.32.5 [825] problem number 5
6.2.32.6 [826] problem number 6
6.2.32.7 [827] problem number 7
6.2.32.8 [828] problem number 8
6.2.32.9 [829] problem number 9
6.2.32.10 [830] problem number 11
6.2.32.11 [831] problem number 12
6.2.32.12 [832] problem number 13
6.2.32.13 [833] problem number 14
6.2.32.14 [834] problem number 15
6.2.32.15 [835] problem number 16
6.2.32.16 [836] problem number 17
6.2.32.17 [837] problem number 18
6.2.32.18 [838] problem number 19
6.2.32.19 [839] problem number 20
6.2.32.20 [840] problem number 21
6.2.32.21 [841] problem number 22
6.2.32.22 [842] problem number 23

6.2.32.1 [821] problem number 1

problem number 821

Added Feb. 7, 2019.

Problem 2.9.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ m x w_x - \left ( n y -x y^k f(x) g(x^n y^m) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  m*x*D[w[x, y], x] - (n*y - x*y^k*f[x]*g[x^n*y^m])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   m*x*diff(w(x,y),x)- ( n*y -x*y^k*f(x)*g(x^n*y^m)  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int _{{\it \_b}}^{x}\!{\frac {1}{g \left ( {{\it \_a}}^{n}{y}^{m} \right ) } \left ( -{{\it \_a}}^{-{\frac {n \left ( k-1 \right ) }{m}}}f \left ( {\it \_a} \right ) g \left ( {{\it \_a}}^{n}{y}^{m} \right ) +{y}^{1-k}{{\it \_a}}^{{\frac {-kn-m+n}{m}}}n \right ) }\,{\rm d}{\it \_a}-\int \!{\frac {1}{g \left ( {x}^{n}{y}^{m} \right ) } \left ( n\int _{{\it \_b}}^{x}\!{\frac {1}{ \left ( g \left ( {{\it \_a}}^{n}{y}^{m} \right ) \right ) ^{2}} \left ( {{\it \_a}}^{{\frac { \left ( n-1 \right ) m-n \left ( k-1 \right ) }{m}}}{y}^{-k+m}m\mbox {D} \left ( g \right ) \left ( {{\it \_a}}^{n}{y}^{m} \right ) +{{\it \_a}}^{{\frac {-kn-m+n}{m}}}{y}^{-k}g \left ( {{\it \_a}}^{n}{y}^{m} \right ) \left ( k-1 \right ) \right ) }\,{\rm d}{\it \_a}g \left ( {x}^{n}{y}^{m} \right ) +{y}^{-k}m{x}^{-{\frac {n \left ( k-1 \right ) }{m}}} \right ) }\,{\rm d}y \right ) \]

____________________________________________________________________________________

6.2.32.2 [822] problem number 2

problem number 822

Added Feb. 9, 2019.

Problem 2.9.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ y^n w_x - \left ( a x^n + g(x) f(y^{n+1} + a x^{n+1}) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  y^n*D[w[x, y], x] - (a*x^n + g[x]*f[y^(n + 1) + a*x^(n + 1)])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  y^n*diff(w(x,y),x)- ( a*x^n + g(x)*f(y^(n+1) + a*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'));
 

sol=()

____________________________________________________________________________________

6.2.32.3 [823] problem number 3

problem number 823

Added Feb. 9, 2019.

Problem 2.9.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( f(\frac {y}{x})+x^\alpha h(\frac {y}{x}) \right ) w_x + \left ( g(\frac {y}{x})+ y x^{\alpha -1} h(\frac {y}{x}) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  (f[y/x] + x^alpha*h[y/x])*D[w[x, y], x] + (g[y/x] + y*x^(alpha - 1)*h[y/x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  (f(y/x)+x^alpha * h(y/x))*diff(w(x,y),x)+ ( g(y/x)+y*x^(alpha-1)*h(y/x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.4 [824] problem number 4

problem number 824

Added Feb. 9, 2019.

Problem 2.9.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( f(a x+b y)+ b x g(a x+b y) \right ) w_x + \left ( h(a x+b y) - a x g(a x+b y) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  (f[a*x + b*y] + b*x*g[a*x + b*y])*D[w[x, y], x] + (h[a*x + b*y] - a*x*g[a*x + b*y])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  (f(a*x+b*y)+b*x*g(a*x+b*y))*diff(w(x,y),x)+ ( h(a*x+b*y)-a*x*g(a*x+b*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.5 [825] problem number 5

problem number 825

Added Feb. 9, 2019.

Problem 2.9.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( f(a x+b y)+ b y g(a x+b y) \right ) w_x + \left ( h(a x+b y) - a y g(a x+b y) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  (f[a*x + b*y] + b*y*g[a*x + b*y])*D[w[x, y], x] + (h[a*x + b*y] - a*y*g[a*x + b*y])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  (f(a*x+b*y)+b*y*g(a*x+b*y))*diff(w(x,y),x)+ ( h(a*x+b*y)-a*y*g(a*x+b*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.6 [826] problem number 6

problem number 826

Added Feb. 9, 2019.

Problem 2.9.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x \left ( f(x^n y^m)+ m x^k g(x^n y^m) \right ) w_x + y \left ( h(x^n y^m) - n x^k g(x^n y^m) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*(f[x^n*y^m] + m*x^k*g[x^n*y^m])*D[w[x, y], x] + y*(h[x^n*y^m] - n*x^k*g[x^n*y^m])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  x*(f(x^n*y^m)+m*x^k*g(x^n*y^m))*diff(w(x,y),x)+ y*( h(x^n*y^m)-n*x^k*g(x^n*y^m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.7 [827] problem number 7

problem number 827

Added Feb. 9, 2019.

Problem 2.9.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x \left ( f(x^n y^m)+ m y^k g(x^n y^m) \right ) w_x + y \left ( h(x^n y^m) - n y^k g(x^n y^m) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*(f[x^n*y^m] + m*y^k*g[x^n*y^m])*D[w[x, y], x] + y*(h[x^n*y^m] - n*y^k*g[x^n*y^m])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  x*(f(x^n*y^m)+m*y^k*g(x^n*y^m))*diff(w(x,y),x)+ y*( h(x^n*y^m)-n*y^k*g(x^n*y^m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.8 [828] problem number 8

problem number 828

Added Feb. 9, 2019.

Problem 2.9.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x \left ( s f(x^n y^m)- m g(x^k y^s) \right ) w_x + y \left (n g(x^k y^s) - k f(x^n y^m) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*(s*f[x^n*y^m] - m*g[x^k*y^s])*D[w[x, y], x] + y*(n*g[x^k*y^s] - k*f[x^n*y^m])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  x*(s*f(x^n*y^m)-m*g(x^k*y^s))*diff(w(x,y),x)+ y*(n*g(x^k*y^s)-k*f(x^n*y^m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.9 [829] problem number 9

problem number 829

Added Feb. 9, 2019.

Problem 2.9.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke 1965.

Solve for \(w(x,y)\)

\[ f_y *w_x - f_x w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[f[x, y], y]*D[w[x, y], x] - D[f[x, y], x]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\{\{w(x,y)\to c_1(\text {InverseFunction}[\text {InverseFunction}[f,2,2],2,2][x,y])\}\}\]

Maple

restart; 
pde :=  diff(f(x,y),y)*diff(w(x,y),x)-diff(f(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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -f \left ( x,y \right ) \right ) \]

____________________________________________________________________________________

6.2.32.10 [830] problem number 11

problem number 830

Added Feb. 9, 2019.

Problem 2.9.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke 1965.

Solve for \(w(x,y)\)

\[ x w_x + \left ( x f(x) g(x^n e^y)- n \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (x*f[x]*g[x^n*Exp[y]] - n)*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)+(x*f(x)*g(x^n*exp(y))-n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( \int _{{\it \_b}}^{y}\! \left ( g \left ( {x}^{n}{{\rm e}^{{\it \_a}}} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}-\int \!f \left ( x \right ) \,{\rm d}x \right ) \]

____________________________________________________________________________________

6.2.32.11 [831] problem number 12

problem number 831

Added Feb. 9, 2019.

Problem 2.9.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke 1965.

Solve for \(w(x,y)\)

\[ m w_x + \left ( m y^k f(x) g(e^{\alpha x} y^m) - \alpha y \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  m*D[w[x, y], x] + (m*y^k*f[x]*g[Exp[alpha*x]*y^m] - alpha*y)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  m*diff(w(x,y),x)+(m*y^k*f(x)*g(exp(alpha*x)*y^m)- alpha*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{m} \left ( -\int \!{\frac {1}{mg \left ( {{\rm e}^{\alpha \,x}}{y}^{m} \right ) } \left ( \alpha \,\int _{{\it \_b}}^{x}\!{\frac {1}{ \left ( g \left ( {{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) \right ) ^{2}} \left ( {{\rm e}^{-{\frac {{\it \_a}\,\alpha \, \left ( k-m-1 \right ) }{m}}}}{y}^{-k+m}\mbox {D} \left ( g \right ) \left ( {{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) m+{{\rm e}^{-{\frac {{\it \_a}\,\alpha \, \left ( k-1 \right ) }{m}}}}{y}^{-k}g \left ( {{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) \left ( k-1 \right ) \right ) }\,{\rm d}{\it \_a}g \left ( {{\rm e}^{\alpha \,x}}{y}^{m} \right ) +{{\rm e}^{-{\frac {\alpha \,x \left ( k-1 \right ) }{m}}}}{y}^{-k}m \right ) }\,{\rm d}ym-\int _{{\it \_b}}^{x}\!{\frac {-mf \left ( {\it \_a} \right ) g \left ( {{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) +{y}^{1-k}\alpha }{g \left ( {{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) }{{\rm e}^{-{\frac {{\it \_a}\,\alpha \, \left ( k-1 \right ) }{m}}}}}\,{\rm d}{\it \_a} \right ) } \right ) \]

____________________________________________________________________________________

6.2.32.12 [832] problem number 13

problem number 832

Added Feb. 9, 2019.

Problem 2.9.3.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left (f(a x+b y)+ b e^{\lambda y} g(a x+b y) \right ) w_x + \left ( h(a x+ b y)- a e^{\lambda y} g(a x + b y) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  (f[a*x + b*y] + b*Exp[lambda*y]*g[a*x + b*y])*D[w[x, y], x] + (h[a*x + b*y] - a*Exp[lambda*y]*g[a*x + b*y])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde := (f(a*x+b*y)+ b*exp(lambda*y)*g(a*x+b*y))*diff(w(x,y),x)+(h(a*x+ b*y)-  a*exp(lambda*y)* g(a*x + b*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.13 [833] problem number 14

problem number 833

Added Feb. 9, 2019.

Problem 2.9.3.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left (f(a x+b y)+ b e^{\lambda x} g(a x+b y) \right ) w_x + \left ( h(a x+ b y)- a e^{\lambda x} g(a x + b y) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  (f[a*x + b*y] + b*Exp[lambda*x]*g[a*x + b*y])*D[w[x, y], x] + (h[a*x + b*y] - a*Exp[lambda*x]*g[a*x + b*y])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde := (f(a*x+b*y)+ b*exp(lambda*x)*g(a*x+b*y))*diff(w(x,y),x)+(h(a*x+ b*y)-  a*exp(lambda*x)* g(a*x + b*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.14 [834] problem number 15

problem number 834

Added Feb. 9, 2019.

Problem 2.9.3.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x \left (f(x^n e^{\alpha y})+\alpha y g(x^n e^{\alpha y}) \right ) w_x + \left ( h(x^n e^{\alpha y})- n y g(x^n e^{\alpha y})) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*(f[x^n*Exp[alpha*y]] + alpha*y*g[x^n*Exp[alpha*y]])*D[w[x, y], x] + (h[x^n*Exp[alpha*y]] - n*y*g[x^n*Exp[alpha*y]])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde := x*(f(x^n*exp(alpha*y))+alpha*y*g(x^n*exp(alpha*y)))*diff(w(x,y),x)+(h(x^n*exp(alpha*y))- n*y*g(x^n*exp(alpha*y)))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.15 [835] problem number 16

problem number 835

Added Feb. 9, 2019.

Problem 2.9.3.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left (f(e^{\alpha x} y^m)+m x g(e^{\alpha x} y^m) \right ) w_x + y \left ( h(e^{\alpha x} y^m)- \alpha x g(e^{\alpha x} y^m)) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  (f[Exp[alpha*x]*y^m] + m*x*g[Exp[alpha*x]*y^m])*D[w[x, y], x] + y*(h[Exp[alpha*x]*y^m] - alpha*x*g[Exp[alpha*x]*y^m])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde := (f(exp(alpha*x)*y^m)+m*x*g(exp(alpha*x)*y^m))*diff(w(x,y),x)+ y*(h(exp(alpha*x)*y^m)- alpha*x*g(exp(alpha*x)*y^m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.16 [836] problem number 17

problem number 836

Added Feb. 9, 2019.

Problem 2.9.3.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( x y f(x) g(x^n \ln y) - n y \ln y \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (x*y*f[x]*g[x^n*Log[y]] - n*y*Log[y])*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)+ (x*y*f(x)*g(x^n*ln(y))-n*y*ln(y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.17 [837] problem number 18

problem number 837

Added Feb. 9, 2019.

Problem 2.9.3.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x\left (f(x^n y^m)+m g(x^n y^m) \ln y\right ) w_x + y \left ( h(x^n y^m) - n g(x^n y^m) \ln y \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*(f[x^n*y^m] + m*g[x^n*y^m]*Log[y])*D[w[x, y], x] + y*(h[x^n*y^m] - n*g[x^n*y^m]*Log[y])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde := x*(f(x^n*y^m)+m*g(x^n*y^m)*ln(y))*diff(w(x,y),x)+ y*(h(x^n*y^m)-n*g(x^n*y^m)*ln(y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.18 [838] problem number 19

problem number 838

Added Feb. 9, 2019.

Problem 2.9.3.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x\left (f(x^n y^m)+m g(x^n y^m) \ln x\right ) w_x + y \left ( h(x^n y^m) - n g(x^n y^m) \ln x \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*(f[x^n*y^m] + m*g[x^n*y^m]*Log[x])*D[w[x, y], x] + y*(h[x^n*y^m] - n*g[x^n*y^m]*Log[x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde := x*(f(x^n*y^m)+m*g(x^n*y^m)*ln(x))*diff(w(x,y),x)+ y*(h(x^n*y^m)-n*g(x^n*y^m)*ln(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.19 [839] problem number 20

problem number 839

Added Feb. 9, 2019.

Problem 2.9.3.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \cos y w_x + \left ( f(x) g(\sin x \sin y) - \cot x \sin y \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  Cos[y]*D[w[x, y], x] + (f[x]*g[Sin[x]*Sin[y]] - Cot[x]*Sin[y])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=cos(y)*diff(w(x,y),x)+ (f(x)* g(sin(x)*sin(y)) - cot(x)*sin(y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.20 [840] problem number 21

problem number 840

Added Feb. 9, 2019.

Problem 2.9.3.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \sin 2x w_x + \left ( \sin 2x \cos ^2 y f(x) g(\tan x \tan y) -\sin 2 y \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  Sin[2*x]*D[w[x, y], x] + (Sin[2*x]*Cos[y]^2*f[x]*g[Tan[x]*Tan[y]] - Sin[2*y])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=sin(2*x)*diff(w(x,y),x)+ (sin(2*x)*cos(y)^2*f(x)*g(tan(x)*tan(y)) -sin(2*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.21 [841] problem number 22

problem number 841

Added Feb. 9, 2019.

Problem 2.9.3.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( x \cos ^2 y f(x) g(x^{2 n} \tan y) - n \sin 2 y \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (x*Cos[y]^2*f[x]*g[x^(2*n)*Tan[y]] - n*Sin[2*y])*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)+ (x *cos(y)^2* f(x)* g(x^(2*n)*tan(y)) - n*sin(2*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.32.22 [842] problem number 23

problem number 842

Added Feb. 9, 2019.

Problem 2.9.3.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( \cos ^2 y f(x) g(e^{2 x} \tan y) -\sin 2 y \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (Cos[y]^2*f[x]*g[Exp[2*x]*Tan[y]] - Sin[2*y])*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)+ (cos(y)^2* f(x)* g(exp(2*x)*tan(y)) -sin(2*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()