Added Jan 1, 2020.
Problem Chapter 8.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + f(x,y) w_z = g(x,y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+f[x,y]*D[w[x,y,z],z]==g[x,y]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^x\frac {g\left (K[2],y+\frac {b (K[2]-x)}{a}\right )}{a}dK[2]\right ) c_1\left (y-\frac {b x}{a},z-\int _1^x\frac {f\left (K[1],y+\frac {b (K[1]-x)}{a}\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ f(x,y)*diff(w(x,y,z),z)=g(x,y)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-\int ^{x}\!{\frac {1}{a}f \left ( {\it \_a},{\frac {ya-b \left ( x-{\it \_a} \right ) }{a}} \right ) }{d{\it \_a}}+z \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a}g \left ( {\it \_a},{\frac {ya-b \left ( x-{\it \_a} \right ) }{a}} \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
Added Jan 1, 2020.
Problem Chapter 8.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + f(x,y) g(z) w_z = h(x,y) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+f[x,y]*g[z]*D[w[x,y,z],z]==h[x,y]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ f(x,y)*g(z)*diff(w(x,y,z),z)=h(x,y)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-\int ^{x}\!f \left ( {\it \_a},{\frac {ya-b \left ( x-{\it \_a} \right ) }{a}} \right ) {d{\it \_a}}+\int \!{\frac {a}{g \left ( z \right ) }}\,{\rm d}z \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a}h \left ( {\it \_a},{\frac {ya-b \left ( x-{\it \_a} \right ) }{a}} \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
Added Jan 1, 2020.
Problem Chapter 8.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ x w_x + y w_y + (z+f(x,y)) w_z = g(x,y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x,y,z],x]+y*D[w[x,y,z],y]+(z+f[x,y])*D[w[x,y,z],z]==g[x,y]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^x\frac {g\left (K[2],\frac {y K[2]}{x}\right )}{K[2]}dK[2]\right ) c_1\left (\frac {y}{x},\frac {z}{x}-\int _1^x\frac {f\left (K[1],\frac {y K[1]}{x}\right )}{K[1]^2}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := x*diff(w(x,y,z),x)+ y*diff(w(x,y,z),y)+ (z+f(x,y))*diff(w(x,y,z),z)=g(x,y)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},{\frac {1}{x} \left ( -\int ^{x}\!{\frac {1}{{{\it \_a}}^{2}}f \left ( {\it \_a},{\frac {{\it \_a}\,y}{x}} \right ) }{d{\it \_a}}x+z \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{{\it \_a}}g \left ( {\it \_a},{\frac {{\it \_a}\,y}{x}} \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
Added Jan 1, 2020.
Problem Chapter 8.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x w_x + b y w_y + f(x,y) g(z) w_z = h(x,y) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*x*D[w[x,y,z],x]+b*y*D[w[x,y,z],y]+f[x,y]*g[z]*D[w[x,y,z],z]==h[x,y]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+ f(x,y)*g(z)*diff(w(x,y,z),z)=h(x,y)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},-\int ^{x}\!{\frac {1}{{\it \_a}}f \left ( {\it \_a},y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) }{d{\it \_a}}+\int \!{\frac {a}{g \left ( z \right ) }}\,{\rm d}z \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a{\it \_a}}h \left ( {\it \_a},y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
Added Jan 1, 2020.
Problem Chapter 8.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+f_2(x) \right ) w_y + \left ( g_1(x,y) z+g_2(x,y) \right ) w_z = h(x,y,z) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(f1[x]*y+f2[x])*D[w[x,y,z],y]+(g1[x,y]*z+g2[x,y])*D[w[x,y,z],z]==h[x,y,z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\int _1^xh\left (K[5],e^{\int _1^{K[5]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[5]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right ),e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[5]}\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3],\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]-\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} \left (z-e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} \int _1^xe^{-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,x\}\right ]dK[3]} \text {g2}\left (K[4],e^{\int _1^{K[4]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[4]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[4]+e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} \int _1^{K[5]}e^{-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\text {InverseFunction}[\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2],1,2]\left [\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^{K[5]}\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]}\right ),\{K[3],1,K[4]\}\right ],\{K[3],1,K[5]\}\right ]}\right ),\{K[3],1,K[5]\}\right ]dK[3]} \text {g2}\left (K[4],e^{\int _1^{K[4]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[4]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )\right )dK[5]} c_1\left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2],e^{-\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]} z-\int _1^xe^{-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (e^{\int _1^x\text {g1}\left (K[3],e^{\int _1^{K[3]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[3]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[3]}\right ),\{K[3],1,x\}\right ]dK[3]} \text {g2}\left (K[4],e^{\int _1^{K[4]}\text {f1}(K[1])dK[1]} \left (e^{-\int _1^x\text {f1}(K[1])dK[1]} y-\int _1^xe^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]+\int _1^{K[4]}e^{-\int _1^{K[2]}\text {f1}(K[1])dK[1]} \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x))*diff(w(x,y,z),y)+ (g__1(x,y)*z+g__2(x,y))*diff(w(x,y,z),z)=h(x,y,z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}},-\int ^{x}\!g_{2} \left ( {\it \_a}, \left ( \int \!f_{2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) {{\rm e}^{-\int \!g_{1} \left ( {\it \_a}, \left ( \int \!f_{2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) \,{\rm d}{\it \_a}}}{d{\it \_a}}+z{{\rm e}^{-\int ^{x}\!g_{1} \left ( {\it \_f}, \left ( \int \!f_{2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}}} \right ) {{\rm e}^{\int ^{x}\!h \left ( {\it \_h}, \left ( \int \!f_{2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}, \left ( \int \!g_{2} \left ( {\it \_h}, \left ( \int \!f_{2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) {{\rm e}^{-\int \!g_{1} \left ( {\it \_h}, \left ( \int \!f_{2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int ^{x}\!g_{2} \left ( {\it \_a}, \left ( \int \!f_{2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) {{\rm e}^{-\int \!g_{1} \left ( {\it \_a}, \left ( \int \!f_{2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) \,{\rm d}{\it \_a}}}{d{\it \_a}}+z{{\rm e}^{-\int ^{x}\!g_{1} \left ( {\it \_f}, \left ( \int \!f_{2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}}} \right ) {{\rm e}^{\int \!g_{1} \left ( {\it \_h}, \left ( \int \!f_{2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h}}} \right ) {d{\it \_h}}}}\]
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Added Jan 1, 2020.
Problem Chapter 8.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+f_2(x) y^k \right ) w_y + \left ( g_1(x,y) z+g_2(x,y) z^m \right ) w_z = h(x,y,z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(f1[x]*y+f2[x]*y^k)*D[w[x,y,z],y]+(g1[x,y]*z+g2[x,y]*z^m)*D[w[x,y,z],z]==h[x,y,z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
$Aborted
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*y^k)*diff(w(x,y,z),y)+ (g__1(x,y)*z+g__2(x,y)*z^m)*diff(w(x,y,z),z)=h(x,y,z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}, \left ( -1+m \right ) \int ^{x}\!{{\rm e}^{\int \!g_{1} \left ( {\it \_h}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}f_{2} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h} \left ( -1+m \right ) }}g_{2} \left ( {\it \_h},{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}f_{2} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_h}}+{z}^{1-m}{{\rm e}^{\int ^{x}\!g_{1} \left ( {\it \_f}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}f_{2} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}} \left ( -1+m \right ) }} \right ) {{\rm e}^{\int ^{x}\!h \left ( {\it \_h}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}f_{2} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}, \left ( {z}^{1-m}{{\rm e}^{\int ^{x}\!g_{1} \left ( {\it \_f}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}f_{2} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}} \left ( -1+m \right ) }}- \left ( -1+m \right ) \left ( \int \!g_{2} \left ( {\it \_h}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}f_{2} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) {{\rm e}^{\int \!g_{1} \left ( {\it \_h}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}f_{2} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h} \left ( -1+m \right ) }}\,{\rm d}{\it \_h}-\int ^{x}\!{{\rm e}^{\int \!g_{1} \left ( {\it \_h}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}f_{2} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h} \left ( -1+m \right ) }}g_{2} \left ( {\it \_h},{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}}}f_{2} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_h}} \right ) \right ) ^{- \left ( -1+m \right ) ^{-1}}{{\rm e}^{\int \!g_{1} \left ( {\it \_h}, \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}f_{2} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h}}} \right ) {d{\it \_h}}}}\]
____________________________________________________________________________________
Added Jan 1, 2020.
Problem Chapter 8.8.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+f_2(x) y^k \right ) w_y + \left ( g_1(x,y) z+g_2(x,y) e^{\lambda z} \right ) w_z = h(x,y,z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(f1[x]*y+f2[x]*y^k)*D[w[x,y,z],y]+(g1[x,y]*z+g2[x,y]*Exp[lambda*z])*D[w[x,y,z],z]==h[x,y,z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*y^k)*diff(w(x,y,z),y)+ (g__1(x,y)*z+g__2(x,y)*exp(lambda*z))*diff(w(x,y,z),z)=h(x,y,z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
____________________________________________________________________________________
Added Jan 1, 2020.
Problem Chapter 8.8.3.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+f_2(x) e^{\lambda y} \right ) w_y + \left ( g_1(x,y) z+g_2(x,y) z^k \right ) w_z = h(x,y,z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(f1[x]*y+f2[x]*Exp[lambda*y])*D[w[x,y,z],y]+(g1[x,y]*z+g2[x,y]*z^k)*D[w[x,y,z],z]==h[x,y,z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g__1(x,y)*z+g__2(x,y)*z^k)*diff(w(x,y,z),z)=h(x,y,z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added Jan 1, 2020.
Problem Chapter 8.8.3.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+f_2(x) e^{\lambda y} \right ) w_y + \left ( g_1(x,y) z+g_2(x,y) e^{\beta z} \right ) w_z = h(x,y,z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(f1[x]*y+f2[x]*Exp[lambda*y])*D[w[x,y,z],y]+(g1[x,y]*z+g2[x,y]*Exp[beta*z])*D[w[x,y,z],z]==h[x,y,z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g__1(x,y)*z+g__2(x,y)*exp(beta*z))*diff(w(x,y,z),z)=h(x,y,z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added Jan 1, 2020.
Problem Chapter 8.8.3.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + \left ( h_1(x,y) +h_2(x,y) z^m \right ) w_z = h_3(x,y,z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = f1[x]*g1[y]*D[w[x,y,z],x]+f2[x]*g2[y]*D[w[x,y,z],y]+(h1[x,y]+h2[x,y]*z^m)*D[w[x,y,z],z]==h3[x,y,z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := f__1(x)*g__1(y)*diff(w(x,y,z),x)+ f__2(x)*g__2(y)*diff(w(x,y,z),y)+ (h__1(x,y)*z+h__2(x,y)*z^m)*diff(w(x,y,z),z)=h__3(x,y,z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y,{\it \_C4} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{f_{1} \left ( {\it \_h} \right ) }h_{3} \left ( {\it \_h},\RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) , \left ( -m\int \!{\frac {1}{f_{1} \left ( {\it \_h} \right ) }h_{2} \left ( {\it \_h},\RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) {{\rm e}^{\int \!{\frac {1}{f_{1} \left ( {\it \_h} \right ) }h_{1} \left ( {\it \_h},\RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \left ( g_{1} \left ( \RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}\,{\rm d}{\it \_h} \left ( -1+m \right ) }} \left ( g_{1} \left ( \RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}\,{\rm d}{\it \_h}+{\it \_C4}+\int \!{\frac {1}{f_{1} \left ( {\it \_h} \right ) }h_{2} \left ( {\it \_h},\RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) {{\rm e}^{\int \!{\frac {1}{f_{1} \left ( {\it \_h} \right ) }h_{1} \left ( {\it \_h},\RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \left ( g_{1} \left ( \RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}\,{\rm d}{\it \_h} \left ( -1+m \right ) }} \left ( g_{1} \left ( \RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}\,{\rm d}{\it \_h} \right ) ^{- \left ( -1+m \right ) ^{-1}}{{\rm e}^{\int \!{\frac {1}{f_{1} \left ( {\it \_h} \right ) }h_{1} \left ( {\it \_h},\RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \left ( g_{1} \left ( \RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}\,{\rm d}{\it \_h}}} \right ) \left ( g_{1} \left ( \RootOf \left ( \int \!{\frac {f_{2} \left ( {\it \_h} \right ) }{f_{1} \left ( {\it \_h} \right ) }}\,{\rm d}{\it \_h}-\int ^{{\it \_Z}}\!{\frac {g_{1} \left ( {\it \_a} \right ) }{g_{2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {g_{1} \left ( y \right ) }{g_{2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}{d{\it \_h}}}}\]
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Added Jan 1, 2020.
Problem Chapter 8.8.3.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + \left ( h_1(x,y) +h_2(x,y) e^{\lambda z} \right ) w_z = h_3(x,y,z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = f1[x]*g1[y]*D[w[x,y,z],x]+f2[x]*g2[y]*D[w[x,y,z],y]+(h1[x,y]+h2[x,y]*Exp[lambda*z])*D[w[x,y,z],z]==h3[x,y,z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed