Added May 31, 2019.
Problem Chapter 6.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + (z+f(x) g(y) ) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + y*D[w[x, y,z], y] +(z+f[x]*g[y])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {z}{x}-\int _1^x\frac {f(K[1]) g\left (\frac {y K[1]}{x}\right )}{K[1]^2}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y,z),x)+ y*diff(w(x,y,z),y)+(z+f(x)*g(y))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y}{x}, \frac {-x \left (\int _{}^{x}\frac {f \left (\mathit {\_a} \right ) g \left (\frac {\mathit {\_a} y}{x}\right )}{\mathit {\_a}^{2}}d\mathit {\_a} \right )+z}{x}\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x +(f_1(x) y +f_2(x) ) w_y + (g_1(y) z +g_2(y) ) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] +(g1[y]*z+g2[y])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}\left (\exp \left (\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (\exp \left (\int _1^x\text {g1}\left (\exp \left (\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[3]\right )\right ),\{K[3],1,x\}\right ]dK[3]\right ) \text {g2}\left (\exp \left (\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[4]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+(g1(y)*z+g2(y))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z \,{\mathrm e}^{-\left (\int _{}^{x}\mathit {g1} \left (\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}}\right )d\mathit {\_f} \right )}-\left (\int _{}^{x}{\mathrm e}^{-\left (\int \mathit {g1} \left (\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d \mathit {\_a} \right )} \mathit {g2} \left (\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}+\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a} \right )} \mathit {f2} \left (\mathit {\_a} \right )d \mathit {\_a} -\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_a} \right )d \mathit {\_a}}\right )d\mathit {\_a} \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x +(y^2-a^2+a \lambda \sinh (\lambda x) -a^2 \sinh ^2(\lambda x)) w_y + f(x) g(z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (y^2-a^2+a*lambda*Sinh[lambda*x]-a^2*Sinh[lambda*x]^2)*D[w[x, y,z], y] +f[x]*g[z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\int _1^z\frac {1}{g(K[2])}dK[2]-\int _1^xf(K[3])dK[3],\frac {2 \lambda e^{\frac {a e^{-\lambda x} \left (e^{2 \lambda x}-1\right )}{\lambda }+\lambda x}}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}-\int _1^{e^{\lambda x}}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (y^2-a^2+a*lambda*sinh(lambda*x)-a^2*sinh(lambda*x)^2)*diff(w(x,y,z),y)+f(x)*g(z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\frac {2 \sqrt {\sinh \left (\lambda x \right )+i}\, \left (-\left (-\frac {\left (\sinh ^{2}\left (\lambda x \right )\right )}{2}+i \sinh \left (\lambda x \right )+\frac {1}{2}\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (i \left (\sinh ^{2}\left (\lambda x \right )\right )+2 \sinh \left (\lambda x \right )-i\right ) \left (a \cosh \left (\lambda x \right )+y \right ) \HeunC \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{-\left (-\sinh \left (\lambda x \right )+i\right ) \left (\sinh ^{2}\left (\lambda x \right )+1\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (2 \left (i \sinh \left (\lambda x \right )+1\right ) \left (\sinh ^{2}\left (\lambda x \right )+1\right ) y +\left (2 i a \left (\sinh ^{3}\left (\lambda x \right )\right )+\left (i \lambda +2 a \right ) \left (\sinh ^{2}\left (\lambda x \right )\right )+2 a -i \lambda +\left (2 i a +2 \lambda \right ) \sinh \left (\lambda x \right )\right ) \cosh \left (\lambda x \right )\right ) \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )}, \int \frac {1}{g \left (z \right )}d z -\left (\int f \left (x \right )d x \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + z^k w_y + g(y) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y,z], x] + z^k*D[w[x, y,z], y] +g[y]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := f(x)*diff(w(x,y,z),x)+ z^k*diff(w(x,y,z),y)+g(y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (z z^{k}+\left (-k -1\right ) \left (\int g \left (y \right )d y \right ), \int \frac {1}{f \left (x \right )}d x -\left (\int _{}^{y}\left (\left (z^{k +1}+\int \left (-k -1\right ) g \left (y \right )d y +\int \left (k +1\right ) g \left (\mathit {\_f} \right )d \mathit {\_f} \right )^{\frac {1}{k +1}}\right )^{-k}d\mathit {\_f} \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + g(y) w_y + h(z) w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\int _1^y\frac {1}{g(K[1])}dK[1]-\int _1^x\frac {1}{f(K[2])}dK[2],\int _1^z\frac {1}{h(K[3])}dK[3]-\int _1^x\frac {1}{f(K[4])}dK[4]\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{g \left (y \right )}d y , -\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{h \left (z \right )}d z \right )\]
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Added May 31, 2019.
Problem Chapter 6.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) w_x + f_2(x) g(y) w_y + f_3(x) h(z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = f1[x]*D[w[x, y,z], x] + f2[x]*g[y]*D[w[x, y,z], y] +f3[x]*g[z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\int _1^y\frac {1}{g(K[1])}dK[1]-\int _1^x\frac {\text {f2}(K[2])}{\text {f1}(K[2])}dK[2],\int _1^z\frac {1}{g(K[3])}dK[3]-\int _1^x\frac {\text {f3}(K[4])}{\text {f1}(K[4])}dK[4]\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\int \frac {1}{g \left (y \right )}d y -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right ), \int \frac {1}{h \left (z \right )}d z -\left (\int \frac {\mathit {f3} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a \sinh (\beta y) w_x + b \sinh (\gamma z) w_y + f_1(x) f_2(z) \sinh (\beta y) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Sinh[beta*y]*D[w[x, y,z], x] + b*Sinh[gamma*z]*D[w[x, y,z], y] +f1[x]*f2[z]*Sinh[beta*y]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\begin {align*} & \left \{w(x,y,z)\to c_1\left (\int _1^z\frac {1}{\text {f2}(K[1])}dK[1]-\int _1^x\frac {\text {f1}(K[2])}{a}dK[2],-\frac {\beta \int _1^x\frac {b \sinh \left (\gamma \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\text {f2}(K[1])}dK[1]\&\right ]\left [-\int _1^x\frac {\text {f1}(K[2])}{a}dK[2]+\int _1^{K[3]}\frac {\text {f1}(K[2])}{a}dK[2]+\int _1^z\frac {1}{\text {f2}(K[1])}dK[1]\right ]\right )}{a}dK[3]+\cosh (\beta y)}{\beta }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\int _1^z\frac {1}{\text {f2}(K[1])}dK[1]-\int _1^x\frac {\text {f1}(K[2])}{a}dK[2],\frac {\cosh (\beta y)}{\beta }-\int _1^x\frac {b \sinh \left (\gamma \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\text {f2}(K[1])}dK[1]\&\right ]\left [-\int _1^x\frac {\text {f1}(K[2])}{a}dK[2]+\int _1^{K[3]}\frac {\text {f1}(K[2])}{a}dK[2]+\int _1^z\frac {1}{\text {f2}(K[1])}dK[1]\right ]\right )}{a}dK[3]\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := a*sinh(beta*y)*diff(w(x,y,z),x)+ b*sinh(gamma1*z)*diff(w(x,y,z),y)+f2(x)*f2(z)*sinh(beta*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\int \frac {a}{\mathit {f2} \left (z \right )}d z -\left (\int \mathit {f2} \left (x \right )d x \right ), \frac {-b \beta \left (\int _{}^{x}\sinh \left (\gamma 1 \RootOf \left (\int \frac {a}{\mathit {f2} \left (z \right )}d z +\int \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} -\left (\int \mathit {f2} \left (x \right )d x \right )-\left (\int _{}^{\mathit {\_Z}}\frac {a}{\mathit {f2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )d\mathit {\_f} \right )+a \cosh \left (\beta y \right )}{b \beta }\right )\]
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