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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (-\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{g \left (y \right )}d y , z -\left (\int \frac {h \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\chi \left (z +\int \frac {h \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g} -\left (\int \frac {h \left (x \right )}{f \left (x \right )}d x \right )\right )+\psi \left (\RootOf \left (\int \frac {1}{f \left (\textit {\_g} \right )}d \textit {\_g} -\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{g \left (y \right )}d y -\left (\int _{}^{\textit {\_Z}}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} \right )\right )\right )+\varphi \left (z +\int \frac {h \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g} -\left (\int \frac {h \left (x \right )}{f \left (x \right )}d x \right )\right )}{f \left (\textit {\_g} \right )}d \textit {\_g}}\]
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (z^{2}-2 \left (\int g \left (y \right )d y \right ), \int \frac {1}{f \left (x \right )}d x -\left (\int _{}^{y}\frac {1}{\sqrt {z^{2}+2 \left (\int g \left (\textit {\_b} \right )d \textit {\_b} \right )-2 \left (\int g \left (y \right )d y \right )}}d \textit {\_b} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {h_{1} \left (\textit {\_g} \right )+h_{2} \left (\RootOf \left (\int \frac {1}{\sqrt {z^{2}+2 \left (\int g \left (\textit {\_g} \right )d \textit {\_g} \right )-2 \left (\int g \left (y \right )d y \right )}}d \textit {\_g} +\int \frac {1}{f \left (x \right )}d x -\left (\int _{}^{y}\frac {1}{\sqrt {z^{2}+2 \left (\int g \left (\textit {\_b} \right )d \textit {\_b} \right )-2 \left (\int g \left (y \right )d y \right )}}d \textit {\_b} \right )-\left (\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_a} \right )}d \textit {\_a} \right )\right )\right )}{\sqrt {z^{2}+2 \left (\int g \left (\textit {\_g} \right )d \textit {\_g} \right )-2 \left (\int g \left (y \right )d y \right )}}d \textit {\_g}}\]
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\int \frac {1}{g \left (y \right )}d y -\left (\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x \right ), \int \frac {1}{h \left (z \right )}d z -\left (\int \frac {f_{3} \left (x \right )}{f_{1} \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {f_{4} \left (x \right )}{f_{1} \left (x \right )}d x}\]
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right ), z \,{\mathrm e}^{-\left (\int g_{1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int g_{1} \left (x \right )d x \right )} g_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int _{}^{x}\left (h_{1} \left (\textit {\_f} \right )+h_{2} \left (\left (y \,{\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int f_{1} \left (\textit {\_f} \right )d \textit {\_f} \right )} f_{2} \left (\textit {\_f} \right )d \textit {\_f} -\left (\int {\mathrm e}^{-\left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int f_{1} \left (\textit {\_f} \right )d \textit {\_f}}\right )\right )d \textit {\_f}}\]
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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=()
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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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right ), \frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int g_{1} \left (x \right )d x \right )} g_{2} \left (x \right )d x \right )-{\mathrm e}^{\left (-z +\int g_{1} \left (x \right )d x \right ) \lambda }}{\lambda }\right ) {\mathrm e}^{\int _{}^{x}\left (h_{1} \left (\textit {\_f} \right )+h_{2} \left (\left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )}+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (\textit {\_f} \right )d \textit {\_f} \right )} f_{2} \left (\textit {\_f} \right )d \textit {\_f} \right )+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int f_{1} \left (x \right )d x \right )} f_{2} \left (x \right )d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\textit {\_f} \right )d \textit {\_f}}\right )\right )d \textit {\_f}}\]
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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=()
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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=()
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