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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},{\frac {1}{x} \left ( -\int ^{x}\!{\frac {f \left ( {\it \_a} \right ) }{{{\it \_a}}^{2}}g \left ( {\frac {{\it \_a}\,y}{x}} \right ) }{d{\it \_a}}x+z \right ) } \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}},-\int ^{x}\!{\it g2} \left ( \left ( \int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) {{\rm e}^{-\int \!{\it g1} \left ( \left ( \int \!{\it f2} \left ( {\it \_a} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}}\,{\rm d}{\it \_a}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}} \right ) \,{\rm d}{\it \_a}}}{d{\it \_a}}+z{{\rm e}^{-\int ^{x}\!{\it g1} \left ( \left ( \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}-\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}} \right ) {{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) {d{\it \_f}}}} \right ) \]
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Added 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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -2\,{\sqrt {\sinh \left ( x\lambda \right ) +i} \left ( \left ( \cosh \left ( x\lambda \right ) a+y \right ) \left ( i \left ( \sinh \left ( x\lambda \right ) \right ) ^{2}-i+2\,\sinh \left ( x\lambda \right ) \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( x\lambda \right ) \right ) -\lambda \,\HeunCPrime \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( x\lambda \right ) \right ) \cosh \left ( x\lambda \right ) \left ( i\sinh \left ( x\lambda \right ) -1/2\, \left ( \sinh \left ( x\lambda \right ) \right ) ^{2}+1/2 \right ) \right ) \left ( \left ( \left ( 2\,i \left ( \sinh \left ( x\lambda \right ) \right ) ^{3}a+ \left ( i\lambda +2\,a \right ) \left ( \sinh \left ( x\lambda \right ) \right ) ^{2}+ \left ( 2\,ia+2\,\lambda \right ) \sinh \left ( x\lambda \right ) -i\lambda +2\,a \right ) \cosh \left ( x\lambda \right ) +2\, \left ( 1+i\sinh \left ( x\lambda \right ) \right ) y \left ( \left ( \sinh \left ( x\lambda \right ) \right ) ^{2}+1 \right ) \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( x\lambda \right ) \right ) - \left ( -\sinh \left ( x\lambda \right ) +i \right ) \lambda \,\cosh \left ( x\lambda \right ) \left ( \left ( \sinh \left ( x\lambda \right ) \right ) ^{2}+1 \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( x\lambda \right ) \right ) \right ) ^{-1}},-\int \!f \left ( x \right ) \,{\rm d}x+\int \! \left ( g \left ( z \right ) \right ) ^{-1}\,{\rm d}z \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( \left ( -k-1 \right ) \int \!g \left ( y \right ) \,{\rm d}y+{z}^{k}z,-\int ^{y}\! \left ( \left ( \left ( 1+k \right ) \int \!g \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+ \left ( -k-1 \right ) \int \!g \left ( y \right ) \,{\rm d}y+{z}^{k}z \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}{d{\it \_f}}+\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y,-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( h \left ( z \right ) \right ) ^{-1}\,{\rm d}z \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y,-\int \!{\frac {{\it f3} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \! \left ( h \left ( z \right ) \right ) ^{-1}\,{\rm d}z \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\it f2} \left ( x \right ) \,{\rm d}x+\int \!{\frac {a}{{\it f2} \left ( z \right ) }}\,{\rm d}z,{\frac {1}{\beta \,b} \left ( -\beta \,b\int ^{x}\!\sinh \left ( \gamma 1\,\RootOf \left ( \int \!{\it f2} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac {a}{{\it f2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\it f2} \left ( x \right ) \,{\rm d}x+\int \!{\frac {a}{{\it f2} \left ( z \right ) }}\,{\rm d}z \right ) \right ) {d{\it \_f}}+a\cosh \left ( \beta \,y \right ) \right ) } \right ) \]
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