Added April 13, 2019.
Problem Chapter 5.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = x f(\frac {y}{x}) w + g(x,y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == x*f[y/x]*w[x,y]+g[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{x f\left (\frac {y}{x}\right )} \left (\int _1^x\frac {e^{-f\left (\frac {y}{x}\right ) K[1]} g\left (K[1],\frac {y K[1]}{x}\right )}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = x*f(y/x)*w(x,y)+g(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{{\it \_a}}g \left ( {\it \_a},{\frac {y{\it \_a}}{x}} \right ) {{\rm e}^{-{\it \_a}\,f \left ( {\frac {y}{x}} \right ) }}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) {{\rm e}^{xf \left ( {\frac {y}{x}} \right ) }}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = f(x,y) w + g(x,y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == f[x,y]*w[x,y]+g[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )}{a K[1]}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )}{a K[1]}dK[1]\right ) g\left (K[2],x^{-\frac {b}{a}} y K[2]^{\frac {b}{a}}\right )}{a K[2]}dK[2]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = f(x,y)*w(x,y)+g(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a{\it \_b}}g \left ( {\it \_b},y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) {{\rm e}^{-{\frac {1}{a}\int \!{\frac {1}{{\it \_b}}f \left ( {\it \_b},y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) }\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{{\it \_a}\,a}f \left ( {\it \_a},y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + g(x) w_y = h(x,y) w + F(x,y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h\left (K[2],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) \left (c_1\left (y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right )+\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {h\left (K[2],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) F\left (K[3],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[3]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[3])}dK[3]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{f \left ( {\it \_f} \right ) }F \left ( {\it \_f},\int \!{\frac {g \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}-\int \!{\frac {g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+y \right ) {{\rm e}^{-\int \!{\frac {1}{f \left ( {\it \_f} \right ) }h \left ( {\it \_f},\int \!{\frac {g \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}-\int \!{\frac {g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+y \right ) }\,{\rm d}{\it \_f}}}}{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+y \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{f \left ( {\it \_b} \right ) }h \left ( {\it \_b},\int \!{\frac {g \left ( {\it \_b} \right ) }{f \left ( {\it \_b} \right ) }}\,{\rm d}{\it \_b}-\int \!{\frac {g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+y \right ) }{d{\it \_b}}}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y + g_0(x)) w_y = h(x,y) w + F(x,y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x])D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h\left (K[3],\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[3])}dK[3]\right ) \left (c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )+\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {h\left (K[3],\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[3])}dK[3]\right ) F\left (K[4],\exp \left (\int _1^{K[4]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[4]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[4])}dK[4]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{f \left ( {\it \_g} \right ) }F \left ( {\it \_g}, \left ( y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}-\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+\int \!{\frac {{\it g0} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}}\,{\rm d}{\it \_g} \right ) {{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \right ) {{\rm e}^{-\int \!{\frac {1}{f \left ( {\it \_g} \right ) }h \left ( {\it \_g}, \left ( y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}-\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+\int \!{\frac {{\it g0} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}}\,{\rm d}{\it \_g} \right ) {{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \right ) }\,{\rm d}{\it \_g}}}}{d{\it \_g}}+{\it \_F1} \left ( -\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{f \left ( {\it \_f} \right ) }h \left ( {\it \_f}, \left ( \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}}\,{\rm d}{\it \_f}-\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) {{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) }{d{\it \_f}}}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y + g_0(x) y^k) w_y = h(x,y) w + F(x,y) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*y^k)D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
$Aborted
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x)*y^k)*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{f \left ( {\it \_g} \right ) }F \left ( {\it \_g}, \left ( \left ( 1-k \right ) \int \!{\frac {{\it g0} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \right ) {{\rm e}^{-\int \!{\frac {1}{f \left ( {\it \_g} \right ) }h \left ( {\it \_g}, \left ( \left ( 1-k \right ) \int \!{\frac {{\it g0} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \right ) }\,{\rm d}{\it \_g}}}}{d{\it \_g}}+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{f \left ( {\it \_f} \right ) }h \left ( {\it \_f}, \left ( \left ( 1-k \right ) \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}}\,{\rm d}{\it \_f}+ \left ( k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) }{d{\it \_f}}}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y + g_0(x) e^{\lambda y}) w_y = h(x,y) w + F(x,y) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*Exp[lambda*y])D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed