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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, z -\left (\int _{}^{x}\frac {f \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )}{a}d\mathit {\_a} \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {g \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )}{a}d\mathit {\_a}}\]
____________________________________________________________________________________
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \int \frac {a}{g \left (z \right )}d z -\left (\int _{}^{x}f \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )d\mathit {\_a} \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\mathit {\_a} , \frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )}{a}d\mathit {\_a}}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y}{x}, \frac {-x \left (\int _{}^{x}\frac {f \left (\mathit {\_a} , \frac {\mathit {\_a} y}{x}\right )}{\mathit {\_a}^{2}}d\mathit {\_a} \right )+z}{x}\right ) {\mathrm e}^{\int _{}^{x}\frac {g \left (\mathit {\_a} , \frac {\mathit {\_a} y}{x}\right )}{\mathit {\_a}}d\mathit {\_a}}\]
____________________________________________________________________________________
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y x^{-\frac {b}{a}}, \int \frac {a}{g \left (z \right )}d z -\left (\int _{}^{x}\frac {f \left (\mathit {\_a} , y \mathit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )}{\mathit {\_a}}d\mathit {\_a} \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\mathit {\_a} , y \mathit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )}{\mathit {\_a} a}d\mathit {\_a}}\]
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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 ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right ), z \,{\mathrm e}^{-\left (\int _{}^{x}g_{1} \left (\mathit {\_f} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} f_{2} \left (\mathit {\_f} \right )d \mathit {\_f} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}-\left (\int _{}^{x}{\mathrm e}^{-\left (\int g_{1} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} f_{2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d \mathit {\_a} \right )} g_{2} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} f_{2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d\mathit {\_a} \right )\right ) {\mathrm e}^{\int _{}^{x}h \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}, \left (z \,{\mathrm e}^{-\left (\int _{}^{x}g_{1} \left (\mathit {\_f} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} f_{2} \left (\mathit {\_f} \right )d \mathit {\_f} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}+\int {\mathrm e}^{-\left (\int g_{1} \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} g_{2} \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} -\left (\int _{}^{x}{\mathrm e}^{-\left (\int g_{1} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} f_{2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d \mathit {\_a} \right )} g_{2} \left (\mathit {\_a} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} f_{2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d\mathit {\_a} \right )\right ) {\mathrm e}^{\int g_{1} \left (\mathit {\_h} , \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h}}\right )d\mathit {\_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 ) = \mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right ), z^{-m +1} {\mathrm e}^{\left (m -1\right ) \left (\int _{}^{x}g_{1} \left (\mathit {\_f} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} f_{2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}+\left (m -1\right ) \left (\int _{}^{x}{\mathrm e}^{\left (m -1\right ) \left (\int g_{1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} g_{2} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} f_{2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d\mathit {\_h} \right )\right ) {\mathrm e}^{\int _{}^{x}h \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}, \left (z^{-m +1} {\mathrm e}^{\left (m -1\right ) \left (\int _{}^{x}g_{1} \left (\mathit {\_f} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} f_{2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}-\left (m -1\right ) \left (\int {\mathrm e}^{\left (m -1\right ) \left (\int g_{1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} g_{2} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} -\left (\int _{}^{x}{\mathrm e}^{\left (m -1\right ) \left (\int g_{1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h} \right )} g_{2} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} f_{2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d\mathit {\_h} \right )\right )\right )^{-\frac {1}{m -1}} {\mathrm e}^{\int g_{1} \left (\mathit {\_h} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} f_{2} \left (\mathit {\_h} \right )d \mathit {\_h} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\mathit {\_h} \right )d \mathit {\_h}}\right )d \mathit {\_h}}\right )d\mathit {\_h}}\]
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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 ) = \mathit {\_F1} \left (-\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y , c_{4}\right ) {\mathrm e}^{\int _{}^{x}\frac {h_{3} \left (\mathit {\_h} , \RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right ), \left (-m \left (\int \frac {{\mathrm e}^{\left (m -1\right ) \left (\int \frac {h_{1} \left (\mathit {\_h} , \RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{f_{1} \left (\mathit {\_h} \right ) g_{1} \left (\RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d \mathit {\_h} \right )} h_{2} \left (\mathit {\_h} , \RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{f_{1} \left (\mathit {\_h} \right ) g_{1} \left (\RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d \mathit {\_h} \right )+c_{4}+\int \frac {{\mathrm e}^{\left (m -1\right ) \left (\int \frac {h_{1} \left (\mathit {\_h} , \RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{f_{1} \left (\mathit {\_h} \right ) g_{1} \left (\RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d \mathit {\_h} \right )} h_{2} \left (\mathit {\_h} , \RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{f_{1} \left (\mathit {\_h} \right ) g_{1} \left (\RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d \mathit {\_h} \right )^{-\frac {1}{m -1}} {\mathrm e}^{\int \frac {h_{1} \left (\mathit {\_h} , \RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{f_{1} \left (\mathit {\_h} \right ) g_{1} \left (\RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d \mathit {\_h}}\right )}{f_{1} \left (\mathit {\_h} \right ) g_{1} \left (\RootOf \left (\int \frac {f_{2} \left (\mathit {\_h} \right )}{f_{1} \left (\mathit {\_h} \right )}d \mathit {\_h} -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right )+\int \frac {g_{1} \left (y \right )}{g_{2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {g_{1} \left (\mathit {\_a} \right )}{g_{2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d\mathit {\_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