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