Added May 31, 2019.
Problem Chapter 7.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c w_z = \alpha x+\beta y+\gamma z + \delta \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*D[w[x,y,z],z]==alpha*x+beta*y+gamma*z+delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {x (a (\alpha x+2 \beta y+2 \delta +2 \gamma z)-x (b \beta +c \gamma ))}{2 a^2}+c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*diff(w(x,y,z),z)= alpha*x+beta*y+gamma*z+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a z -c x}{a}\right )+\frac {\left (\beta y +\gamma z +\delta \right ) x}{a}+\frac {\left (a \alpha -b \beta -c \gamma \right ) x^{2}}{2 a^{2}}\]
____________________________________________________________________________________
Added May 31, 2019.
Problem Chapter 7.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a z w_y + b y w_z = c x+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]==c*x+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )+\frac {c x^2}{2}+s x\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= c*x+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{y}-\frac {\left (-\ln \left (\frac {\textit {\_a} a b +\sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )+\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )\right ) c +\left (-c x -s \right ) \sqrt {a b}}{\sqrt {a b}\, \sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_a} +\textit {\_F1} \left (\frac {a \,z^{2}-b \,y^{2}}{a}, -\frac {-\sqrt {a b}\, x +\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\]
____________________________________________________________________________________
Added May 31, 2019.
Problem Chapter 7.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x + a_0) w_y + (b_1 x+b_0) w_z = \alpha x+\beta y+\gamma z + \delta \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*x+a0)*D[w[x, y,z], y] +(b1*x+b0)*D[w[x,y,z],z]==alpha*x+beta*y+gamma*z+delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {1}{6} x \left (-3 \text {a0} \beta x-2 \text {a1} \beta x^2+3 \alpha x-3 \text {b0} \gamma x-2 \text {b1} \gamma x^2+6 \beta y+6 \delta +6 \gamma z\right )+c_1\left (-\text {a0} x-\frac {\text {a1} x^2}{2}+y,-\text {b0} x-\frac {\text {b1} x^2}{2}+z\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a1*x+a0)*diff(w(x,y,z),y)+(b1*x+b0)*diff(w(x,y,z),z)=alpha*x+beta*y+gamma*z+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (-2 \mathit {a1} \beta -2 \gamma \mathit {b1} \right ) x^{3}}{6}+\frac {\left (-3 \beta \mathit {a0} -3 \gamma \mathit {b0} +3 \alpha \right ) x^{2}}{6}+\frac {\left (6 \beta y +6 \gamma z +6 \delta \right ) x}{6}+\textit {\_F1} \left (-\frac {1}{2} \mathit {a1} \,x^{2}-\mathit {a0} x +y , -\frac {1}{2} \mathit {b1} \,x^{2}-\mathit {b0} x +z \right )\]
____________________________________________________________________________________
Added May 31, 2019.
Problem Chapter 7.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_2 y + a_1 x + a_0) w_y + (b_2 y+b_1 x + b_0) w_z = c_2 y+c_1 z+c_0 x+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a2*y+a1*x+a0)*D[w[x, y,z], y] +(b2*y+b1*x+b0)*D[w[x,y,z],z]==c2*y+c1*z+c0*x+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {6 \text {a2}^4 c_1\left (\frac {e^{-\text {a2} x} (\text {a2} (\text {a0}+\text {a2} y)+\text {a1} \text {a2} x+\text {a1})}{\text {a2}^2},\frac {e^{-\text {a2} x} \left (\text {a2} \left (2 \text {a0} \text {b2} \left (\text {a2} x e^{\text {a2} x}+1\right )-\text {a2} \left (\text {a2} e^{\text {a2} x} \left (2 \text {b0} x+\text {b1} x^2-2 z\right )+2 \text {b2} y \left (e^{\text {a2} x}-1\right )\right )\right )+\text {a1} \text {b2} \left (\text {a2}^2 x^2 e^{\text {a2} x}+2 \text {a2} x+2\right )\right )}{2 \text {a2}^3}\right )+\text {a2} \left (3 \text {a0} \text {b2} \text {c1} \left (\text {a2}^2 x^2-2 \text {a2} x+2\right )-6 \text {a0} \text {a2} \text {c2} (\text {a2} x-1)+\text {a2}^3 x \left (-3 \text {b0} \text {c1} x-2 \text {b1} \text {c1} x^2+3 \text {c0} x+6 \text {c1} z+6 s\right )+6 \text {a2}^2 y (\text {c2}-\text {b2} \text {c1} x)+6 \text {a2} \text {b2} \text {c1} y\right )+\text {a1} \left (\text {b2} \text {c1} \left (2 \text {a2}^3 x^3-3 \text {a2}^2 x^2+6\right )-3 \text {a2} \text {c2} \left (\text {a2}^2 x^2-2\right )\right )}{6 \text {a2}^4}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a2*y+a1*x+a0)*diff(w(x,y,z),y)+(b2*y+b1*x+b0)*diff(w(x,y,z),z)=c2*y+c1*z+c0*x+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {6 \left (-\frac {\mathit {b1} \mathit {c1} \,x^{2}}{3}+\mathit {c1} z +s +\left (-\frac {\mathit {b0} \mathit {c1}}{2}+\frac {\mathit {c0}}{2}\right ) x \right ) \mathit {a2}^{4} x +6 \mathit {a2}^{4} \textit {\_F1} \left (\frac {\left (\mathit {a2}^{2} y +\mathit {a1} +\left (\mathit {a1} x +\mathit {a0} \right ) \mathit {a2} \right ) {\mathrm e}^{-\mathit {a2} x}}{\mathit {a2}^{2}}, \frac {-2 \mathit {a0} \mathit {a2} \mathit {b2} +\left (-\mathit {b1} \,x^{2}-2 \mathit {b0} x +2 z \right ) \mathit {a2}^{3}+\left (\mathit {a1} \,x^{2}+2 \mathit {a0} x -2 y \right ) \mathit {a2}^{2} \mathit {b2} -2 \mathit {a1} \mathit {b2}}{2 \mathit {a2}^{3}}\right )+6 \mathit {a1} \mathit {b2} \mathit {c1} +\left (2 \mathit {a1} \mathit {b2} \mathit {c1} \,x^{3}+6 \mathit {c2} y +\left (3 \mathit {a0} \mathit {b2} \mathit {c1} -3 \mathit {c2} \mathit {a1} \right ) x^{2}+\left (-6 \mathit {b2} \mathit {c1} y -6 \mathit {c2} \mathit {a0} \right ) x \right ) \mathit {a2}^{3}+\left (-3 \mathit {c1} \mathit {a1} \mathit {b2} \,x^{2}-6 \mathit {c1} \mathit {a0} \mathit {b2} x +6 \mathit {b2} \mathit {c1} y +6 \mathit {c2} \mathit {a0} \right ) \mathit {a2}^{2}+\left (6 \mathit {a0} \mathit {b2} \mathit {c1} +6 \mathit {c2} \mathit {a1} \right ) \mathit {a2}}{6 \mathit {a2}^{4}}\]
____________________________________________________________________________________
Added May 31, 2019.
Problem Chapter 7.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y + k_1 x + k_0) w_y + (b z+s_1 x + s_0) w_z = c_1 x+c_0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+k1*x+k0)*D[w[x, y,z], y] +(b*z+s1*x+s0)*D[w[x,y,z],z]==c1*x+c0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-a x} \left (a^2 y+a (\text {k0}+\text {k1} x)+\text {k1}\right )}{a^2},\frac {e^{-b x} \left (b^2 z+b (\text {s0}+\text {s1} x)+\text {s1}\right )}{b^2}\right )+\text {c0} x+\frac {\text {c1} x^2}{2}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a*y+k1*x+k0)*diff(w(x,y,z),y)+(b*z+s1*x+s0)*diff(w(x,y,z),z)=c1*x+c0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\mathit {c1} \,x^{2}}{2}+\mathit {c0} x +\textit {\_F1} \left (\frac {\left (a^{2} y +\left (\mathit {k1} x +\mathit {k0} \right ) a +\mathit {k1} \right ) {\mathrm e}^{-a x}}{a^{2}}, \frac {\left (b^{2} z +\left (\mathit {s1} x +\mathit {s0} \right ) b +\mathit {s1} \right ) {\mathrm e}^{-b x}}{b^{2}}\right )\]
____________________________________________________________________________________
Added May 31, 2019.
Problem Chapter 7.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b y w_y + c z w_z = \alpha x+\beta y+\gamma z + \delta \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +c*z*D[w[x,y,z],z]==alpha*x+beta*y+gamma*z+delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-\frac {b x}{a}},z e^{-\frac {c x}{a}}\right )+\frac {x (\alpha x+2 \delta )}{2 a}+\frac {\beta y}{b}+\frac {\gamma z}{c}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+c*z*diff(w(x,y,z),z)=alpha*x+beta*y+gamma*z+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\alpha \,x^{2}}{2 a}+\frac {\delta x}{a}+\frac {\beta y}{b}+\frac {\gamma z}{c}+\textit {\_F1} \left (y \,{\mathrm e}^{-\frac {b x}{a}}, z \,{\mathrm e}^{-\frac {c x}{a}}\right )\]
Hand solution
Solve \[ aw_{x}+byw_{y}+czw_{z}=\alpha x+\beta y+\gamma z+\delta \] The parametrization invariant Lagrange-Charpit equations are\[ \frac {dx}{a}=\frac {dy}{by}=\frac {dz}{cz}=\frac {dw}{\alpha x+\beta y+\gamma z+\delta }\] Solving \(\frac {dx}{a}=\frac {dy}{by}\) gives\begin {align} \frac {b}{a}dx & =\frac {dy}{y}\nonumber \\ \frac {b}{a}x & =\ln y+C_{1}\tag {1}\\ \ln y & =\frac {b}{a}x-C_{1}\nonumber \\ y & =C_{1}e^{\frac {b}{a}x}\nonumber \\ C_{1} & =ye^{-\frac {b}{a}x}\nonumber \end {align}
Equation \(\frac {dx}{a}=\frac {dz}{cz}\) gives\begin {align} \frac {c}{a}dx & =\frac {dz}{z}\nonumber \\ \frac {c}{a}x & =\ln z+C_{2}\nonumber \\ \ln z & =\frac {c}{a}x-C_{2}\nonumber \\ z & =C_{2}e^{\frac {c}{a}x}\nonumber \\ C_{2} & =ze^{-\frac {c}{a}x}\tag {2} \end {align}
And \(\frac {dx}{a}=\frac {dw}{\alpha x+\beta y+\gamma z+\delta }\) gives\begin {align*} \frac {\alpha x+\beta y+\gamma z+\delta }{a}dx & =dw\\ \left ( \frac {\alpha }{a}x+\beta \frac {y}{a}+\gamma \frac {z}{a}+\frac {\delta }{a}\right ) dx & =dw \end {align*}
But from (1) \(y=C_{1}e^{\frac {b}{a}x}\) and from (2) \(z=C_{2}e^{\frac {c}{a}x}\). Hence the above becomes\[ \left ( \frac {\alpha }{a}x+\frac {\beta }{a}C_{1}e^{\frac {b}{a}x}+\frac {\gamma }{a}C_{2}e^{\frac {c}{a}x}+\frac {\delta }{a}\right ) dx=dw \] Integrating\[ \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}C_{1}e^{\frac {b}{a}x}+\frac {\gamma }{c}C_{2}e^{\frac {c}{a}x}+\frac {\delta }{a}x=w+C_{3}\] But \(C_{2}=ze^{-\frac {c}{a}x}\) and \(C_{1}=ye^{-\frac {b}{a}x}\), hence the above becomes\begin {align*} \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}ye^{-\frac {b}{a}x}e^{\frac {b}{a}x}+\frac {\gamma }{c}ze^{-\frac {c}{a}x}e^{\frac {c}{a}x}+\frac {\delta }{a}x & =w+C_{3}\\ \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x & =w+C_{3}\\ C_{3} & =\left ( \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x\right ) -w \end {align*}
Since \(C_{3}=F\left ( C_{1},C_{2}\right ) \) then the solution is\begin {align*} \left ( \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x\right ) -w & =F\left ( ye^{-\frac {b}{a}x},ze^{-\frac {c}{a}x}\right ) \\ w\left ( x,y,z\right ) & =F\left ( ye^{-\frac {b}{a}x},ze^{-\frac {c}{a}x}\right ) +\left ( \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x\right ) \end {align*}
(sign change on \(F\) does not matter, since arbitrary function, can be renamed).
____________________________________________________________________________________
Added June 1, 2019.
Problem Chapter 7.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + a z w_y + b y w_z = c \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + a*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]==c; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c \log (x)+c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := x*diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)=c; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {c \ln \left (\frac {a b y}{\sqrt {a b}}+\sqrt {a^{2} z^{2}}\right )+\sqrt {a b}\, \textit {\_F1} \left (\frac {a \,z^{2}-b \,y^{2}}{a}, x \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right )}{\sqrt {a b}}\]
Hand solution
Solve \[ xw_{x}+azw_{y}+byw_{z}=c \] The parametrization invariant Lagrange-Charpit equations are\[ \frac {dx}{x}=\frac {dy}{az}=\frac {dz}{by}=\frac {dw}{c}\] Solving \(\frac {dy}{az}=\frac {dz}{by}\) gives\begin {align} \frac {b}{a}ydy & =zdz\nonumber \\ \frac {b}{a}y^{2} & =z^{2}+C_{1}\tag {1}\\ C_{1} & =\frac {b}{a}y^{2}-z^{2}\nonumber \\ & =\frac {by^{2}-az^{2}}{a}\nonumber \end {align}
Equation \(\frac {dx}{x}=\frac {dy}{az}\) gives\[ a\frac {dx}{x}=\frac {dy}{z}\] But from (1) \(z=\sqrt {\frac {b}{a}y^{2}-C_{1}}\), hence the above becomes\[ a\frac {dx}{x}=\frac {dy}{\sqrt {\frac {b}{a}y^{2}-C_{1}}}\] Integrating gives\begin {align*} a\ln x & =\sqrt {\frac {a}{b}}\ln \left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) +C_{2}\\ \ln x & =\sqrt {\frac {1}{ab}}\ln \left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) +\frac {C_{2}}{a} \end {align*}
Let \(\frac {C_{2}}{a}=C_{3}\) and the above becomes\begin {align*} x & =C_{3}\left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) ^{\sqrt {\frac {1}{ab}}}\\ C_{3} & =x\left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) ^{-\sqrt {\frac {1}{ab}}} \end {align*}
But from (1) \begin {equation} C_{1}=\frac {b}{a}y^{2}-z^{2}\tag {2} \end {equation} Hence \(C_{3}\) simplifies to\begin {align} C_{3} & =x\left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-\left ( \frac {b}{a}y^{2}-z^{2}\right ) }\right ) ^{-\sqrt {\frac {1}{ab}}}\nonumber \\ & =x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\tag {4} \end {align}
And \(\frac {dx}{x}=\frac {dw}{c}\) gives\[ \ln x=\frac {1}{c}w+C_{4}\] But \(C_{4}=F\left ( C_{1},C_{3}\right ) \). Hence \begin {align*} \ln x-\frac {1}{c}w & =F\left ( \frac {b}{a}y^{2}-z^{2},x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\right ) \\ -\frac {1}{c}w & =F\left ( \frac {b}{a}y^{2}-z^{2},x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\right ) -\ln x\\ w\left ( x,y,z\right ) & =cF\left ( \frac {b}{a}y^{2}-z^{2},x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\right ) +c\ln x \end {align*}
Verified OK under the assumptions that \(a>0,b>0,z>0\).
____________________________________________________________________________________
Added June 1, 2019.
Problem Chapter 7.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a b x w_x + b(a y+b z) w_y + a(a y-b z) w_z = c \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*b*x*D[w[x, y,z], x] + b*(a*y+b*z)*D[w[x, y,z], y] +a*(a*y-b*z)*D[w[x,y,z],z]==c; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*b*x*diff(w(x,y,z),x)+ b*(a*y+b*z)*diff(w(x,y,z),y)+a*(a*y-b*z)*diff(w(x,y,z),z)=c; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {2 \sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}\, b \textit {\_F1} \left (-\frac {1}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}, x \left (\frac {\frac {\sqrt {2}\, a^{2} y}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}+\left (\frac {a y}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}+\frac {b z}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}\right ) \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}{\sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, a}{2 \sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}}\right )+\sqrt {2}\, c \ln \left (\frac {\sqrt {2}\, a^{2} y}{\left (-a^{2} y^{2}+2 a b y z +b^{2} z^{2}\right ) \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}+\frac {a y}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}+\frac {b z}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}\right )}{2 \sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}\, b}\]
____________________________________________________________________________________
Added June 1, 2019.
Problem Chapter 7.2.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (a_1 x+a_0) w_x + (b_1 y+b_0) w_y + (c_1 z+c_0) w_z = \alpha x+\beta y+\gamma z + \delta \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a1*x+a0)*D[w[x, y,z], x] + (b1*y+b0)*D[w[x, y,z], y] +(c1*z+c0)*D[w[x,y,z],z]==alpha*x+beta*y+gamma*z+delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {(\text {b0}+\text {b1} y) (\text {a0}+\text {a1} x)^{-\frac {\text {b1}}{\text {a1}}}}{\text {b1}},\frac {(\text {c0}+\text {c1} z) (\text {a0}+\text {a1} x)^{-\frac {\text {c1}}{\text {a1}}}}{\text {c1}}\right )-\frac {\text {a0} \alpha \log (\text {a0}+\text {a1} x)}{\text {a1}^2}+\frac {\log (\text {a0}+\text {a1} x) (-\text {b0} \beta \text {c1}-\text {b1} \text {c0} \gamma +\text {b1} \text {c1} \delta )}{\text {a1} \text {b1} \text {c1}}+\frac {\alpha x}{\text {a1}}+\frac {\text {b0} \beta }{\text {b1}^2}+\frac {\beta y}{\text {b1}}+\frac {\gamma (\text {c0}+\text {c1} z)}{\text {c1}^2}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := (a1*x+a0)*diff(w(x,y,z),x)+ (b1*y+b0)*diff(w(x,y,z),y)+(c1*z+c0)*diff(w(x,y,z),z)=alpha*x+beta*y+gamma*z+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\mathit {a1}^{2} \mathit {b1}^{2} \mathit {c1}^{2} \textit {\_F1} \left (\frac {\left (\mathit {b1} y +\mathit {b0} \right ) \left (\mathit {a1} x +\mathit {a0} \right )^{-\frac {\mathit {b1}}{\mathit {a1}}}}{\mathit {b1}}, \frac {\left (\mathit {c1} z +\mathit {c0} \right ) \left (\mathit {a1} x +\mathit {a0} \right )^{-\frac {\mathit {c1}}{\mathit {a1}}}}{\mathit {c1}}\right )-\left (\mathit {a0} \alpha \mathit {b1} \mathit {c1} +\left (\mathit {b0} \beta \mathit {c1} +\left (\gamma \mathit {c0} -\mathit {c1} \delta \right ) \mathit {b1} \right ) \mathit {a1} \right ) \mathit {b1} \mathit {c1} \ln \left (\mathit {a1} x +\mathit {a0} \right )+\left (\alpha \,\mathit {b1}^{2} \mathit {c1}^{2} x +\left (\mathit {b1} \beta \,\mathit {c1}^{2} y +\mathit {b0} \beta \,\mathit {c1}^{2}+\left (\mathit {c1} z +\mathit {c0} \right ) \mathit {b1}^{2} \gamma \right ) \mathit {a1} \right ) \mathit {a1}}{\mathit {a1}^{2} \mathit {b1}^{2} \mathit {c1}^{2}}\]
____________________________________________________________________________________