Added December 1, 2019.
Problem Chapter 8.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ f(x) w_x + g(y) w_y + h(z) w_z = \left ( \varphi (z)+\psi (y)+\chi (z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x,y,z],x]+g[y]*D[w[x,y,z],y]+h[z]*D[w[x,y,z],z]==(varphi[z]+psi[y]+chi[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(x)*diff(w(x,y,z),x)+ g(y)*diff(w(x,y,z),y)+ h(x)*diff(w(x,y,z),z)=(varphi(z)+psi(y)+chi(z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y,-\int \!{\frac {h \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+z \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{f \left ( {\it \_g} \right ) } \left ( \varphi \left ( \int \!{\frac {h \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}-\int \!{\frac {h \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+z \right ) +\psi \left ( \RootOf \left ( \int \! \left ( f \left ( {\it \_g} \right ) \right ) ^{-1}\,{\rm d}{\it \_g}-\int ^{{\it \_Z}}\! \left ( g \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \right ) +\chi \left ( \int \!{\frac {h \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}-\int \!{\frac {h \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+z \right ) \right ) }{d{\it \_g}}}}\]
____________________________________________________________________________________
Added December 1, 2019.
Problem Chapter 8.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ f(x) w_x + z w_y + g(y) w_z = \left ( h_2(x)+h_1(y) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x,y,z],x]+z*D[w[x,y,z],y]+g[y]*D[w[x,y,z],z]==(h2[x]+h1[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 := f(x)*diff(w(x,y,z),x)+ z*diff(w(x,y,z),y)+ g(y)*diff(w(x,y,z),z)=(h__2(x)+h__1(y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y,-\int ^{y}\!{\frac {1}{\sqrt {2\,\int \!g \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}{d{\it \_b}}+\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x \right ) {{\rm e}^{\int ^{y}\!{ \left ( h_{2} \left ( \RootOf \left ( \int \!{\frac {1}{\sqrt {2\,\int \!g \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}\,{\rm d}{\it \_g}-\int ^{{\it \_Z}}\! \left ( f \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-\int ^{y}\!{\frac {1}{\sqrt {2\,\int \!g \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}{d{\it \_b}}+\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x \right ) \right ) +h_{1} \left ( {\it \_g} \right ) \right ) {\frac {1}{\sqrt {2\,\int \!g \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+{z}^{2}-2\,\int \!g \left ( y \right ) \,{\rm d}y}}}}{d{\it \_g}}}}\]
____________________________________________________________________________________
Added December 1, 2019.
Problem Chapter 8.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ f_1(x) w_x + f_2(x) g(y) w_y + f_3(x) h(z) w_z = f_4(x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = f1[x]*D[w[x,y,z],x]+f2[x]*g[y]*D[w[x,y,z],y]+f3[x]*h[z]*D[w[x,y,z],z]==f4[x]*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)*diff(w(x,y,z),x)+ f__2(x)*g(y)*diff(w(x,y,z),y)+ f__3(x)*h(z)*diff(w(x,y,z),z)=f__4(x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\frac {f_{2} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y,-\int \!{\frac {f_{3} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x+\int \! \left ( h \left ( z \right ) \right ) ^{-1}\,{\rm d}z \right ) {{\rm e}^{\int \!{\frac {f_{4} \left ( x \right ) }{f_{1} \left ( x \right ) }}\,{\rm d}x}}\]
____________________________________________________________________________________
Added December 1, 2019.
Problem Chapter 8.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left (f_1(x) y+f_2(x) \right ) w_y + \left (g_1(x) z+g_2(y) \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(f1[x]*y+f2[x])*D[w[x,y,z],y]+(g1[x]*z+g2[x])*D[w[x,y,z],z]==(h1[x]+h2[y])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[5]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )\right )dK[5]\right ) c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right \}\right \}\]
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)*z+g__2(x))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}},-\int \!g_{2} \left ( x \right ) {{\rm e}^{-\int \!g_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+z{{\rm e}^{-\int \!g_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int ^{x}\!h_{1} \left ( {\it \_f} \right ) +h_{2} \left ( \left ( \int \!f_{2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}-\int \!f_{2} \left ( x \right ) {{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}}}\]
____________________________________________________________________________________
Added December 1, 2019.
Problem Chapter 8.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left (f_1(x) y+f_2(x) y^k\right ) w_y + \left (g_1(y) z+g_2(x) z^m \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(f1[x]*y+f2[x]*y^k)*D[w[x,y,z],y]+(g1[y]*z+g2[x])*D[w[x,y,z],z]==(h1[x]+h2[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\left (\text {h1}(K[5])+\text {h2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[5]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )\right )dK[5]\right ) c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),z \exp \left (-\int _1^x\text {g1}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[3]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (\exp \left (\int _1^x\text {g1}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[3]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[3]\right )\right ),\{K[3],1,x\}\right ]dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right \}\right \}\]
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+ (f__1(y)*y+f__2(x)*y^k)*diff(w(x,y,z),y)+ (g__1(y)*z+g__2(x))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added December 1, 2019.
Problem Chapter 8.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left (f_1(x) y+f_2(x) y^k\right ) w_y + \left (g_1(x)+g_2(y) e^{\lambda z} \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]
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]+g2[x]*Exp[lambda*z])*D[w[x,y,z],z]==(h1[x]+h2[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 := diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*y^k)*diff(w(x,y,z),y)+ (g__1(x)+g__2(x)*exp(lambda*z))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}},{\frac {-\lambda \,\int \!g_{2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!g_{1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( z-\int \!g_{1} \left ( x \right ) \,{\rm d}x \right ) }}}{\lambda }} \right ) {{\rm e}^{\int ^{x}\!h_{1} \left ( {\it \_f} \right ) +h_{2} \left ( \left ( \left ( -k+1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}f_{2} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+ \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}}f_{2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!f_{1} \left ( x \right ) \,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!f_{1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}}}\]
____________________________________________________________________________________
Added December 1, 2019.
Problem Chapter 8.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left (f_1(x) y+f_2(x) e^{\lambda y}\right ) w_y + \left (g_1(y) z+g_2(x) z^k \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]
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[y]*z+g2[x]*z^k)*D[w[x,y,z],z]==(h1[x]+h2[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 := diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g__1(y)*z+g__2(x)*z^k)*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added December 1, 2019.
Problem Chapter 8.8.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + \left (f_1(x) y+f_2(x) e^{\lambda y}\right ) w_y + \left (g_1(x)+g_2(x) e^{\beta z} \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]
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]+g2[y]*Exp[beta*z])*D[w[x,y,z],z]==(h1[x]+h2[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 := 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)+g__2(y)*exp(beta*z))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________