____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f(x) y^2 -f(x) g(x) y+ g'(x)\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f]; pde = D[w[x, y], x] + (f[x]*y^2 - f[x]*g[x]*y + Derivative[1][g][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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+( f(x)*y^2 -f(x)*g(x)*y+ diff(g(x),x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x - \left ( f'(x) y^2 -f(x) g(x) y+ g(x)\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f]; pde = D[w[x, y], x] - (Derivative[1][f][x]*y^2 - f[x]*g[x]*y + g[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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)-( diff(f(x),x)*y^2 -f(x)*g(x)*y+ g(x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{f \left ( x \right ) y-1} \left ( yf \left ( x \right ) \int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{\int \!f \left ( x \right ) g \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x-f \left ( x \right ) {{\rm e}^{-\int \!{\frac {- \left ( f \left ( x \right ) \right ) ^{2}g \left ( x \right ) +2\,{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}-\int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{\int \!f \left ( x \right ) g \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x \right ) } \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( g(x)(y-f(x))^2 + f'(x)\right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f]; pde = D[w[x, y], x] + (g[x]*(y - f[x])^2 + Derivative[1][f][x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-f(x) \int _1^x g(K[2]) \, dK[2]+y \int _1^x g(K[2]) \, dK[2]+1}{y-f(x)}\right )\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+(g(x)*(y-f(x))^2 + diff(f(x),x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {\int \!g \left ( x \right ) \,{\rm d}xy-f \left ( x \right ) \int \!g \left ( x \right ) \,{\rm d}x+1}{y-f \left ( x \right ) }} \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \frac {f'(x)}{g(x)} y^2 - \frac {g'(x)}{f(x)}\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f]; pde = D[w[x, y], x] + ((Derivative[1][f][x]*y^2)/g[x] - Derivative[1][g][x]/f[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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+(diff(f(x),x)/g(x)* y^2 - diff(g(x),x)/f(x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{ \left ( f \left ( x \right ) y+g \left ( x \right ) \right ) f \left ( x \right ) } \left ( y \left ( f \left ( x \right ) \right ) ^{2}\int \!{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{ \left ( f \left ( x \right ) \right ) ^{2}g \left ( x \right ) }}\,{\rm d}x+g \left ( x \right ) f \left ( x \right ) \int \!{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{ \left ( f \left ( x \right ) \right ) ^{2}g \left ( x \right ) }}\,{\rm d}x+1 \right ) } \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f^2(x) w_x + \left ( f'(x) y^2 -g(x) (y-f(x)) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f]; pde = f[x]^2*D[w[x, y], x] + (Derivative[1][f][x]*y^2 - g[x]*(y - f[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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := f(x)^2*diff(w(x,y),x)+(diff(f(x),x)*y^2 -g(x)*(y-f(x)) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2 - \frac {f''(x)}{f(x)} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f]; pde = D[w[x, y], x] + (y^2 - Derivative[2][f][x]/f[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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+(y^2 - diff(f(x),x,x)/f(x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {y \left ( f \left ( x \right ) \right ) ^{2}\int \! \left ( f \left ( x \right ) \right ) ^{-2}\,{\rm d}x+\int \! \left ( f \left ( x \right ) \right ) ^{-2}\,{\rm d}x \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) f \left ( x \right ) +1}{f \left ( x \right ) \left ( f \left ( x \right ) y+{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) }} \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ g(x) w_x + \left ( a f(x) g(x) y^3 + (b f(x) g^3(x) + g'(x)) y+ c f(x) g^4(x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f]; pde = g[x]*D[w[x, y], x] + (a*f[x]*g[x]*y^3 + (b*f[x]*g[x]^3 + Derivative[1][g][x])*y + c*f[x]*g[x]^4)*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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := g(x)*diff(w(x,y),x)+(a*f(x)*g(x)*y^3 + (b*f(x)*g(x)^3 + diff(g(x),x))*y+ c*f(x)*g(x)^4)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {b}^{3}\sum _{{\it \_R}=\RootOf \left ( {{\it \_Z}}^{3}a{c}^{2}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac {{\it \_R}\,cg \left ( x \right ) +by}{cg \left ( x \right ) }} \right ) }-b\int \! \left ( g \left ( x \right ) \right ) ^{2}f \left ( x \right ) \,{\rm d}x \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f(x) y^3+3 f(x) h(x) y^2+(g(x)+3 f(x) h^2(x)) y+ f(x) h^3(x) + g(x) h(x) - h'(x) \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; pde = D[w[x, y], x] + (f[x]*y^3 + 3*f[x]*h[x]*y^2 + (g[x] + 3*f[x]*h[x]^2)*y + f[x]*h[x]^3 + g[x]*h[x] - Derivative[1][h][x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {4 y h(x) \left (\int _1^x f(K[2]) e^{2 \text {Integrate}[g(K[1]),\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}]} \, dK[2]\right )+2 h(x)^2 \int _1^x f(K[2]) e^{2 \text {Integrate}[g(K[1]),\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}]} \, dK[2]+2 y^2 \left (\int _1^x f(K[2]) e^{2 \text {Integrate}[g(K[1]),\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}]} \, dK[2]\right )+e^{2 \int _1^x g(K[1]) \, dK[1]}}{(h(x)+y)^2}\right )\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+(f(x)*y^3+3*f(x)*h(x)*y^2+(g(x)+3*f(x)*h(x)^2)*y+ f(x)*h(x)^3 + g(x)* h(x) - diff(h(x),x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {2\,{y}^{2}\int \!f \left ( x \right ) {{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+4\,yh \left ( x \right ) \int \!f \left ( x \right ) {{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+2\, \left ( h \left ( x \right ) \right ) ^{2}\int \!f \left ( x \right ) {{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}}{ \left ( y+h \left ( x \right ) \right ) ^{2}}} \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \frac {g'(x)}{f^2(x) (a g(x) + b)^3} y^3 + \frac {f'(x)}{f(x)} y + f(x) g'(x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; pde = D[w[x, y], x] + ((Derivative[1][g][x]*y^3)/(f[x]^2*(a*g[x] + b)^3) + (Derivative[1][f][x]*y)/f[x] + f[x]*Derivative[1][g][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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+(diff(g(x),x)/(f(x)^2 *(a*g(x)+b)^3)*y^3 + diff(f(x),x)/f(x) * y + f(x)*diff(g(x),x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {a}^{3}\sum _{{\it \_R}=\RootOf \left ( -{a}^{3}{\it \_Z}+{{\it \_Z}}^{3}+{a}^{3} \right ) }{\frac {1}{-{a}^{3}+3\,{{\it \_R}}^{2}}\ln \left ( {\frac {-g \left ( x \right ) f \left ( x \right ) {\it \_R}\,a-f \left ( x \right ) {\it \_R}\,b+ya}{ \left ( ag \left ( x \right ) +b \right ) f \left ( x \right ) }} \right ) }-\ln \left ( ag \left ( x \right ) +b \right ) \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.6.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( (y-f(x))(y-g(x)) \left (y- \frac {a f(x)+b g(x)}{a+b} \right ) h(x) + \frac {y-g(x)}{f(x)-g(x)} f'(x) + \frac {y-f(x)}{g(x)-f(x)} g'(x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; pde = D[w[x, y], x] + ((y - f[x])*(y - g[x])*(y - (a*f[x] + b*g[x])/(a + b))*h[x] + ((y - g[x])*Derivative[1][f][x])/(f[x] - g[x]) + ((y - f[x])*Derivative[1][g][x])/(g[x] - f[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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+((y-f(x))*(y-g(x))*(y- (a*f(x)+b*g(x))/(a+b))*h(x)+(y-g(x))/(f(x)-g(x))*diff(f(x),x)+ (y-f(x))/(g(x)-f(x))*diff(g(x),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 ( 1/3\,{\frac {1}{{a}^{2}+2\,ab+{b}^{2}} \left ( -{\frac {a}{b \left ( a+b \right ) } \left ( \ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {a}^{3}+2\,\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {a}^{2}b+2\,\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) a{b}^{2}+\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {b}^{3}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) {a}^{3}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) {a}^{2}b-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) {a}^{2}b-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) {b}^{3} \right ) }-2\,{\frac {1}{a+b} \left ( \ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {a}^{3}+2\,\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {a}^{2}b+2\,\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) a{b}^{2}+\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {b}^{3}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) {a}^{3}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) {a}^{2}b-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) {a}^{2}b-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) {b}^{3} \right ) }-{\frac {b}{a \left ( a+b \right ) } \left ( \ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {a}^{3}+2\,\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {a}^{2}b+2\,\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) a{b}^{2}+\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ) {b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) {b}^{3}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) {a}^{3}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) {a}^{2}b-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ) {b}^{2}-f \left ( x \right ) {b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) {a}^{2}b-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac {{a}^{3}y+2\,y{a}^{2}b+2\,ya{b}^{2}+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ) {b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) {b}^{3} \right ) }+2\,{a}^{2}\int \!h \left ( x \right ) f \left ( x \right ) g \left ( x \right ) \,{\rm d}x-{a}^{2}\int \! \left ( f \left ( x \right ) \right ) ^{2}h \left ( x \right ) \,{\rm d}x-{a}^{2}\int \! \left ( g \left ( x \right ) \right ) ^{2}h \left ( x \right ) \,{\rm d}x+2\,ab\int \!h \left ( x \right ) f \left ( x \right ) g \left ( x \right ) \,{\rm d}x-ab\int \! \left ( f \left ( x \right ) \right ) ^{2}h \left ( x \right ) \,{\rm d}x-ab\int \! \left ( g \left ( x \right ) \right ) ^{2}h \left ( x \right ) \,{\rm d}x+2\,{b}^{2}\int \!h \left ( x \right ) f \left ( x \right ) g \left ( x \right ) \,{\rm d}x-{b}^{2}\int \! \left ( f \left ( x \right ) \right ) ^{2}h \left ( x \right ) \,{\rm d}x-{b}^{2}\int \! \left ( g \left ( x \right ) \right ) ^{2}h \left ( x \right ) \,{\rm d}x \right ) } \right ) \]
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Added Feb. 7, 2019.
Problem 2.8.6.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f(x) y^2 + g'(x) y+ a f(x) e^{2 g(x)} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; pde = D[w[x, y], x] + (f[x]*y^2 + Derivative[1][g][x]*y + a*f[x]*Exp[2*g[x]])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> a > 0], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y e^{-g(x)}}{\sqrt {a}}\right )-\sqrt {a} \int _1^x f(K[1]) e^{g(K[1])} \, dK[1]\right )\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+(f(x)*y^2 + diff(g(x),x)*y+ a*f(x)*exp(2*g(x)) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming a>0 ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( \sqrt {a}\int \!f \left ( x \right ) {{\rm e}^{g \left ( x \right ) }}\,{\rm d}x-\arctan \left ( {\frac {{{\rm e}^{-g \left ( x \right ) }}y}{\sqrt {a}}} \right ) \right ) \]
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Added Feb. 7, 2019.
Problem 2.8.6.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f'(x) y^2+ a e^{\lambda x} f(x) y+a e^{\lambda x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; pde = D[w[x, y], x] + (Derivative[1][f][x]*y^2 + a*Exp[lambda*x]*f[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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+(diff(f(x),x)*y^2+ a*exp(lambda*x)* f(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{f \left ( x \right ) y+1} \left ( yf \left ( x \right ) \int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x+f \left ( x \right ) {{\rm e}^{\int \!{\frac {{{\rm e}^{\lambda \,x}} \left ( f \left ( x \right ) \right ) ^{2}a-2\,{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}+\int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x \right ) } \right ) \]