Added June 27, 2019.
Problem Chapter 7.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + g(y) w_y + h(z) w_z = \Phi (x) + \Psi (x) + \chi (x) \]
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]== phi[x]+psi[x]+chi[x]; 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(z)*diff(w(x,y,z),z)= phi(x)+psi(x)+chi(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int \frac {\chi \left (x \right )+\phi \left (x \right )+\psi \left (x \right )}{f \left (x \right )}d x +\mathit {\_F1} \left (-\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{g \left (y \right )}d y , -\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{h \left (z \right )}d z \right )\]
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Added June 27, 2019.
Problem Chapter 7.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + z w_y + g(y) w_z = h_2(x)+h_1(y) \]
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]; 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)= h2(x)+h1(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{y}\frac {\mathit {h1} \left (\mathit {\_g} \right )+\mathit {h2} \left (\RootOf \left (\int \frac {1}{\sqrt {z^{2}+2 \left (\int g \left (\mathit {\_g} \right )d \mathit {\_g} \right )-2 \left (\int g \left (y \right )d y \right )}}d \mathit {\_g} +\int \frac {1}{f \left (x \right )}d x -\left (\int _{}^{y}\frac {1}{\sqrt {z^{2}+2 \left (\int g \left (\mathit {\_b} \right )d \mathit {\_b} \right )-2 \left (\int g \left (y \right )d y \right )}}d\mathit {\_b} \right )-\left (\int _{}^{\mathit {\_Z}}\frac {1}{f \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{\sqrt {z^{2}+2 \left (\int g \left (\mathit {\_g} \right )d \mathit {\_g} \right )-2 \left (\int g \left (y \right )d y \right )}}d\mathit {\_g} +\mathit {\_F1} \left (z^{2}-2 \left (\int g \left (y \right )d y \right ), \int \frac {1}{f \left (x \right )}d x -\left (\int _{}^{y}\frac {1}{\sqrt {z^{2}+2 \left (\int g \left (\mathit {\_b} \right )d \mathit {\_b} \right )-2 \left (\int g \left (y \right )d y \right )}}d\mathit {\_b} \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) w_x + f_2(x) g(y) w_y + f_3(x) h(z) w_z = f_4(x) \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := f1(x)*diff(w(x,y,z),x)+ f2(x)*g(y)*diff(w(x,y,z),y)+ f3(x)*h(z)*diff(w(x,y,z),z)= f4(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int \frac {\mathit {f4} \left (x \right )}{\mathit {f1} \left (x \right )}d x +\mathit {\_F1} \left (\int \frac {1}{g \left (y \right )}d y -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right ), \int \frac {1}{h \left (z \right )}d z -\left (\int \frac {\mathit {f3} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x)) w_y + (g_1(x) z+g_2(y)) w_z = h_1(x)+h_2(y) \]
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[y])*D[w[x,y,z],z]== h1[x]+h2[y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[5]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )\right )dK[5]+c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}\left (\exp \left (\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[4]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+ (g1(x)*z+g2(y))*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\mathit {h1} \left (\mathit {\_f} \right )+\mathit {h2} \left (\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )\right )d\mathit {\_f} +\mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z \,{\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )}-\left (\int _{}^{x}{\mathrm e}^{-\left (\int \mathit {g1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {g2} \left (\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}}\right )d\mathit {\_g} \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(y) z^m) w_z = h_1(x)+h_2(y) \]
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]*z+g2[y]*z^m)*D[w[x,y,z],z]== h1[x]+h2[y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[5]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )\right )dK[5]+c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[4]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )\right \}\right \}\]
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)*z+g2(y)*z^m)*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\mathit {h1} \left (\mathit {\_f} \right )+\mathit {h2} \left (\left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )\right )d\mathit {\_f} +\mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z^{-m +1} {\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )}+\left (m -1\right ) \left (\int _{}^{x}{\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {g2} \left (\left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )+\left (-k +1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}}\right )d\mathit {\_g} \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(y) e^{\lambda z}) w_z = h_1(x)+h_2(y) \]
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]*z+g2[y]*Exp[lambda*z])*D[w[x,y,z],z]== h1[x]+h2[y]; 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)*z+g2(y)*exp(lambda*z))*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added June 27, 2019.
Problem Chapter 7.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g_1(x) z+g_2(y) z^k) w_z = h_1(x)+h_2(y) \]
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]*z+g2[y]*z^k)*D[w[x,y,z],z]== h1[x]+h2[y]; 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)*z+g2(y)*z^k)*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\mathit {h1} \left (\mathit {\_f} \right )+\mathit {h2} \left (\frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )\right )d\mathit {\_f} +\mathit {\_F1} \left (\frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )-{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}{\lambda }, z^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int _{}^{x}{\mathrm e}^{\left (k -1\right ) \left (\int \mathit {g1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {g2} \left (\frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_g} \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g_1(x) +g_2(y) e^{\beta z}) w_z = h_1(x)+h_2(y) \]
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]+g2[y]*Exp[beta*z])*D[w[x,y,z],z]== h1[x]+h2[y]; 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)+g2(y)*exp(beta*z))*diff(w(x,y,z),z)= h1(x)+h2(y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\mathit {h1} \left (\mathit {\_f} \right )+\mathit {h2} \left (\frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )\right )d\mathit {\_f} +\mathit {\_F1} \left (\frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )-{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}{\lambda }, \frac {-\beta \left (\int _{}^{x}{\mathrm e}^{\beta \left (\int \mathit {g1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {g2} \left (\frac {\lambda \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )+\ln \left (\frac {1}{\left (-\left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right )+\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) \lambda +{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}\right )}{\lambda }\right )d\mathit {\_g} \right )-{\mathrm e}^{\left (-z +\int \mathit {g1} \left (x \right )d x \right ) \beta }}{\beta }\right )\]
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