Added July 1, 2019.
Problem Chapter 8.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = x y z w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] +b*D[w[x,y,z],z]== x*y*z*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {1}{12} x^2 (a x (b x-2 z)-2 b x y+6 y z)\right ) c_1(y-a x,z-b x)\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*diff(w(x,y,z),y)+b*diff(w(x,y,z),z)= x*y*z*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\frac {\left (a b \,x^{2}+6 y z +\left (-2 a z -2 b y \right ) x \right ) x^{2}}{12}}\]
____________________________________________________________________________________
Added July 1, 2019.
Problem Chapter 8.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c w_z = (k x^3+ s y^2) w \]
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]== (k*x^3+s*y^2)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {x \left (3 a^2 \left (k x^3+4 s y^2\right )-12 a b s x y+4 b^2 s x^2\right )}{12 a^3}\right ) c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )\right \}\right \}\]
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)= (k*x^3+s*y^2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a z -c x}{a}\right ) {\mathrm e}^{\frac {\left (a^{2} k \,x^{3}+4 a^{2} s \,y^{2}-4 a b s x y +\frac {4}{3} b^{2} s \,x^{2}\right ) x}{4 a^{3}}}\]
____________________________________________________________________________________
Added July 1, 2019.
Problem Chapter 8.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b y w_y + c z w_z = (k x+ s \sqrt x) w \]
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]== (k*x+s*Sqrt[x])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {3 k x^2+4 s x^{3/2}}{6 a}} c_1\left (y e^{-\frac {b x}{a}},z e^{-\frac {c x}{a}}\right )\right \}\right \}\]
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)= (k*x+s*sqrt(x))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (y \,{\mathrm e}^{-\frac {b x}{a}}, z \,{\mathrm e}^{-\frac {c x}{a}}\right ) {\mathrm e}^{\frac {3 k \,x^{2}+4 s \,x^{\frac {3}{2}}}{6 a}}\]
____________________________________________________________________________________
Added July 1, 2019.
Problem Chapter 8.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a z w_y + b y w_z = (c \sqrt x + s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + b*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]== (c*Sqrt[x]+s)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {2}{3} c x^{3/2}+s x} c_1\left (\frac {1}{2} e^{-b x} \left (y \left (-e^{2 b x}\right )+z e^{2 b x}+y+z\right ),\frac {1}{2} e^{-b x} \left (y e^{2 b x}-z e^{2 b x}+y+z\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+b*z*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= (c*sqrt(x)+s)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (-y^{2}+z^{2}, \frac {b x -\ln \left (y +z \right )}{b}\right ) {\mathrm e}^{\int _{}^{y}\frac {\sqrt {\frac {b x +\ln \left (\textit {\_a} +\sqrt {\textit {\_a}^{2}-y^{2}+z^{2}}\right )-\ln \left (y +z \right )}{b}}\, c +s}{\sqrt {\textit {\_a}^{2}-y^{2}+z^{2}}\, b}d \textit {\_a}}\]
____________________________________________________________________________________
Added July 1, 2019.
Problem Chapter 8.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x^2 w_x + b y^2 w_y + c z^2 w_z = k x y z w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*D[w[x, y,z], x] + b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== k*x*y*z*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right ) \exp \left (\frac {k x y z \left (b y (a x-c z) \log \left (\frac {a x}{y}\right )+c z (b y-a x) \log \left (\frac {a x}{z}\right )\right )}{(a x-b y) (a x-c z) (b y-c z)}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x^2*diff(w(x,y,z),x)+b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)= k*x*y*z*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\frac {a x}{y}\right )^{\frac {b k x \,y^{2} z}{\left (b y -c z \right ) \left (a x -b y \right )}} \left (\frac {a x}{z}\right )^{-\frac {c k x y \,z^{2}}{\left (b y -c z \right ) \left (a x -c z \right )}} \textit {\_F1} \left (\frac {a x -b y}{a x y}, \frac {a x -c z}{a x z}\right )\]
____________________________________________________________________________________