Added June 27, 2019.
Problem Chapter 7.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + f(x,y) w_z = g(x,y) \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {g\left (K[2],y+\frac {b (K[2]-x)}{a}\right )}{a}dK[2]+c_1\left (y-\frac {b x}{a},z-\int _1^x\frac {f\left (K[1],y+\frac {b (K[1]-x)}{a}\right )}{a}dK[1]\right )\right \}\right \}\] Kernel message inconsistent or redundant transcendental equation
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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {1}{a}g \left ( {\it \_a},{\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\!{\frac {1}{a}f \left ( {\it \_a},{\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) }{d{\it \_a}}+z \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + f(x,y) g(z) w_z = h(x,y) \]
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]; 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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {1}{a}h \left ( {\it \_a},{\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\!f \left ( {\it \_a},{\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) {d{\it \_a}}+\int \!{\frac {a}{g \left ( z \right ) }}\,{\rm d}z \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + (z+f(x,y)) w_z = g(x,y) \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {g\left (K[2],\frac {y K[2]}{x}\right )}{K[2]}dK[2]+c_1\left (\frac {y}{x},\frac {z}{x}-\int _1^x\frac {f\left (K[1],\frac {y K[1]}{x}\right )}{K[1]^2}dK[1]\right )\right \}\right \}\]
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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {1}{{\it \_a}}g \left ( {\it \_a},{\frac {{\it \_a}\,y}{x}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}},{\frac {1}{x} \left ( -\int ^{x}\!{\frac {1}{{{\it \_a}}^{2}}f \left ( {\it \_a},{\frac {{\it \_a}\,y}{x}} \right ) }{d{\it \_a}}x+z \right ) } \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + f(x,y) g(z) w_z = h(x,y) \]
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]; 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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {1}{{\it \_a}\,a}h \left ( {\it \_a},y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},-\int ^{x}\!{\frac {1}{{\it \_a}}f \left ( {\it \_a},y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) }{d{\it \_a}}+\int \!{\frac {a}{g \left ( z \right ) }}\,{\rm d}z \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)y+f_2(x)) w_y + (g_1(x,y) z+ g_2(x,y)) w_z = h(x,y,z) \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to 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 )+\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]\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+ (g1(x,y)*z+g2(x,y))*diff(w(x,y,z),z)= h(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 ) =\int ^{x}\!h \left ( {\it \_h}, \left ( \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}, \left ( \int \!{\it g2} \left ( {\it \_h}, \left ( \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_h}, \left ( \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int ^{x}\!{\it g2} \left ( {\it \_a}, \left ( \int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_a}, \left ( \int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) \,{\rm d}{\it \_a}}}{d{\it \_a}}+z{{\rm e}^{-\int ^{x}\!{\it g1} \left ( {\it \_f}, \left ( \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}}} \right ) {{\rm e}^{\int \!{\it g1} \left ( {\it \_h}, \left ( \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \right ) \,{\rm d}{\it \_h}}} \right ) {d{\it \_h}}+{\it \_F1} \left ( -\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}},-\int ^{x}\!{\it g2} \left ( {\it \_a}, \left ( \int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_a}, \left ( \int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) \,{\rm d}{\it \_a}}}{d{\it \_a}}+z{{\rm e}^{-\int ^{x}\!{\it g1} \left ( {\it \_f}, \left ( \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}}} \right ) \]
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Added June 27, 2019.
Problem Chapter 7.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)y+f_2(x) y^k) w_y + (g_1(x,y) z+ g_2(x,y) z^m) w_z = h(x,y,z) \]
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]; 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)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x,y)*z+g2(x,y)*z^m)*diff(w(x,y,z),z)= h(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 ) =\int ^{x}\!h \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}, \left ( {z}^{1-m}{{\rm e}^{\int ^{x}\!{\it g1} \left ( {\it \_f},{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f} \left ( k-1 \right ) }}\,{\rm d}{\it \_f}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_f}} \left ( m-1 \right ) }}+ \left ( m-1 \right ) \left ( \int ^{x}\!{{\rm e}^{\int \!{\it g1} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_h} \left ( m-1 \right ) }}{\it g2} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g} \left ( k-1 \right ) }}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_h}}-\int \!{\it g2} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {{\rm e}^{\int \!{\it g1} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_h} \left ( m-1 \right ) }}\,{\rm d}{\it \_h} \right ) \right ) ^{- \left ( m-1 \right ) ^{-1}}{{\rm e}^{\int \!{\it g1} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_h}}} \right ) {d{\it \_h}}+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}, \left ( m-1 \right ) \int ^{x}\!{{\rm e}^{\int \!{\it g1} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_h} \left ( m-1 \right ) }}{\it g2} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g} \left ( k-1 \right ) }}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_h}}+{z}^{1-m}{{\rm e}^{\int ^{x}\!{\it g1} \left ( {\it \_f},{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f} \left ( k-1 \right ) }}\,{\rm d}{\it \_f}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_f}} \left ( m-1 \right ) }} \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)y+f_2(x) y^k) w_y + (g_1(x,y)+ g_2(x,y) e^{\lambda z}) w_z = h(x,y,z) \]
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]+g2[x,y]*Exp[lambda*z])*D[w[x,y,z],z]== h[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)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x,y)+g2(x,y)*exp(lambda*z))*diff(w(x,y,z),z)= h(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 ) =\int ^{x}\!h \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}},{\frac {1}{\lambda } \left ( \ln \left ( \left ( \left ( -\int \!{\it g2} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {{\rm e}^{\lambda \,\int \!{\it g1} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}+\int ^{x}\!{\it g2} \left ( {\it \_b},{{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g} \left ( k-1 \right ) }}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {{\rm e}^{\lambda \,\int \!{\it g1} \left ( {\it \_b},{{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_b} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b} \left ( k-1 \right ) }}\,{\rm d}{\it \_b}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_b}}}{d{\it \_b}} \right ) \lambda +{{\rm e}^{\lambda \, \left ( -z+\int ^{x}\!{\it g1} \left ( {\it \_b},{{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_b} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b} \left ( k-1 \right ) }}\,{\rm d}{\it \_b}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_b}} \right ) }} \right ) ^{-1} \right ) +\lambda \,\int \!{\it g1} \left ( {\it \_h},{{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_h} \right ) } \right ) {d{\it \_h}}+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}},{\frac {1}{\lambda } \left ( -\lambda \,\int ^{x}\!{\it g2} \left ( {\it \_b},{{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_g} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_g} \right ) \,{\rm d}{\it \_g} \left ( k-1 \right ) }}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {{\rm e}^{\lambda \,\int \!{\it g1} \left ( {\it \_b},{{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_b} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b} \left ( k-1 \right ) }}\,{\rm d}{\it \_b}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) \,{\rm d}{\it \_b}}}{d{\it \_b}}-{{\rm e}^{\lambda \, \left ( -z+\int ^{x}\!{\it g1} \left ( {\it \_b},{{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}}} \left ( \left ( -k+1 \right ) \int \!{\it f2} \left ( {\it \_b} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_b} \right ) \,{\rm d}{\it \_b} \left ( k-1 \right ) }}\,{\rm d}{\it \_b}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) {d{\it \_b}} \right ) }} \right ) } \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)+f_2(x) e^{\lambda y}) w_y + (g_1(x,y) z+ g_2(x,y) z^k) w_z = h(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]+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]; 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)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g1(x,y)*z+g2(x,y)*z^k)*diff(w(x,y,z),z)= h(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 ) =\int ^{x}\!h \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }},{{\rm e}^{{\frac {1}{k-1} \left ( -\ln \left ( \left ( k-1 \right ) \int ^{x}\!{\it g2} \left ( {\it \_f},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}{\lambda }} \right ) {{\rm e}^{\int \!{\it g1} \left ( {\it \_f},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}{\lambda }} \right ) \,{\rm d}{\it \_f} \left ( k-1 \right ) }}{d{\it \_f}}+ \left ( -k+1 \right ) \int \!{\it g2} \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }} \right ) {{\rm e}^{\int \!{\it g1} \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) }}\,{\rm d}{\it \_h}+{z}^{-k+1}{{\rm e}^{\int ^{x}\!{\it g1} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) {d{\it \_a}} \left ( k-1 \right ) }} \right ) +\int \!{\it g1} \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }} \right ) \,{\rm d}{\it \_h} \left ( k-1 \right ) \right ) }}} \right ) {d{\it \_h}}+{\it \_F1} \left ( {\frac {-\lambda \,\int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}}{\lambda }}, \left ( k-1 \right ) \int ^{x}\!{\it g2} \left ( {\it \_f},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}{\lambda }} \right ) {{\rm e}^{\int \!{\it g1} \left ( {\it \_f},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}{\lambda }} \right ) \,{\rm d}{\it \_f} \left ( k-1 \right ) }}{d{\it \_f}}+{z}^{-k+1}{{\rm e}^{\int ^{x}\!{\it g1} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) {d{\it \_a}} \left ( k-1 \right ) }} \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)+f_2(x) e^{\lambda y}) w_y + (g_1(x,y)+ g_2(x,y) e^{\beta z}) w_z = h(x,y,z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] + (g1[x,y]+g2[x,y]*Exp[beta*z])*D[w[x,y,z],z]== h[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)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g1(x,y)+g2(x,y)*exp(beta*z))*diff(w(x,y,z),z)= h(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 ) =\int ^{x}\!h \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }},{\frac {1}{\beta } \left ( \ln \left ( \left ( -\int \!{\it g2} \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }} \right ) {{\rm e}^{\beta \,\int \!{\it g1} \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h}\beta +\beta \,\int ^{x}\!{\it g2} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) {{\rm e}^{\beta \,\int \!{\it g1} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) \,{\rm d}{\it \_a}}}{d{\it \_a}}+{{\rm e}^{\beta \, \left ( -z+\int ^{x}\!{\it g1} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) {d{\it \_a}} \right ) }} \right ) ^{-1} \right ) +\beta \,\int \!{\it g1} \left ( {\it \_h},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_h} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}}\,{\rm d}{\it \_h} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_h} \right ) \,{\rm d}{\it \_h}}{\lambda }} \right ) \,{\rm d}{\it \_h} \right ) } \right ) {d{\it \_h}}+{\it \_F1} \left ( {\frac {-\lambda \,\int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}}{\lambda }},{\frac {1}{\beta } \left ( -\beta \,\int ^{x}\!{\it g2} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) {{\rm e}^{\beta \,\int \!{\it g1} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) \,{\rm d}{\it \_a}}}{d{\it \_a}}-{{\rm e}^{\beta \, \left ( -z+\int ^{x}\!{\it g1} \left ( {\it \_a},{\frac {\ln \left ( \left ( {{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}+\lambda \, \left ( \int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a} \right ) \right ) ^{-1} \right ) +\lambda \,\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{\lambda }} \right ) {d{\it \_a}} \right ) }} \right ) } \right ) \]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + (h_1(x,y)+ h_2(x,y) z^m) w_z = h_3(x,y,z) \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; local gamma; pde := f1(x)*g1(y)*diff(w(x,y,z),x)+ f2(x)*g2(y)*diff(w(x,y,z),y)+ (h1(x,y)+h2(x,y)*z^m)*diff(w(x,y,z),z)= h3(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 7.8.3.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + (h_1(x,y)+ h_2(x,y) e^{\lambda z}) w_z = h_3(x,y,z) \]
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*x])*D[w[x,y,z],z]== h3[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed