Added Jan 6, 2020.
Problem Chapter 9.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c w_z = (\alpha x + \beta ) w + p x + q \]
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)*w[x,y,z]+p*x+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {x (\alpha x+2 \beta )}{2 a}} c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\alpha x+\beta )^2}{2 a \alpha }} (\alpha q-\beta p) \text {erf}\left (\frac {\alpha x+\beta }{\sqrt {2} \sqrt {a} \sqrt {\alpha }}\right )}{\sqrt {a} \alpha ^{3/2}}-\frac {p}{\alpha }\right \}\right \}\]
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)*w(x,y,z)+p*x+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (-\alpha q +\beta p \right ) \erf \left (\frac {\sqrt {2}\, \sqrt {\frac {\alpha }{a}}\, x}{2}+\frac {\sqrt {2}\, \beta }{2 \sqrt {\frac {\alpha }{a}}\, a}\right ) {\mathrm e}^{\frac {\beta ^{2}}{2 a \alpha }}}{2}+\sqrt {\frac {\alpha }{a}}\, \left (\alpha \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {a z -c x}{a}\right )-p \,{\mathrm e}^{-\frac {\left (\alpha x +2 \beta \right ) x}{2 a}}\right ) a \right ) {\mathrm e}^{\frac {\left (\alpha x +2 \beta \right ) x}{2 a}}}{\sqrt {\frac {\alpha }{a}}\, a \alpha }\]
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Added Jan 6, 2020.
Problem Chapter 9.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a z w_y + b y w_z = (c x + k) w + p x + q \]
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+k)*w[x,y,z]+p*x+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (c x+2 k)} c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(c x+k)^2}{2 c}} \text {erf}\left (\frac {c x+k}{\sqrt {2} \sqrt {c}}\right ) (c q-k p)}{c^{3/2}}-\frac {p}{c}\right \}\right \}\]
Maple ✓
restart; local gamma; 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+k)*w(x,y,z)+p*x+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int _{}^{y}\frac {\left (\left (\ln \left (\frac {\mathit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )\right ) p +\sqrt {a b}\, \left (p x +q \right )\right ) {\mathrm e}^{-\frac {\int \frac {\left (\ln \left (\frac {\mathit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )\right ) c +\sqrt {a b}\, \left (c x +k \right )}{\sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}d \mathit {\_b}}{\sqrt {a b}}}}{\sqrt {a b}\, \sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}d\mathit {\_b} +\mathit {\_F1} \left (\frac {a z^{2}-b y^{2}}{a}, -\frac {-\sqrt {a b}\, x +\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (\ln \left (\frac {\mathit {\_a} a b +\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )\right ) c +\sqrt {a b}\, \left (c x +k \right )}{\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}d\mathit {\_a}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 x+a_0) w_y + (b_1 x+b_0) w_z = (c_1 x + c_0) w + s_1 x + s_0 \]
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]==(c1*x+c0)*w[x,y,z]+s1*x+s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (-\text {a0} x-\frac {\text {a1} x^2}{2}+y,-\text {b0} x-\frac {\text {b1} x^2}{2}+z\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*x+a__0)*diff(w(x,y,z),y)+ (b__1*x+b__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x+s__0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (2 c_{1}^{\frac {5}{2}} \mathit {\_F1} \left (-\frac {1}{2} a_{1} x^{2}-a_{0} x +y , -\frac {1}{2} b_{1} x^{2}-b_{0} x +z \right )-2 c_{1}^{\frac {3}{2}} s_{1} \,{\mathrm e}^{-\frac {1}{2} c_{1} x^{2}-c_{0} x}+\sqrt {2}\, \sqrt {\pi }\, \left (-c_{0} s_{1} +c_{1} s_{0} \right ) c_{1} \erf \left (\frac {\sqrt {2}\, \left (\sqrt {c_{1}}\, x +\frac {c_{0}}{\sqrt {c_{1}}}\right )}{2}\right ) {\mathrm e}^{\frac {c_{0}^{2}}{2 c_{1}}}\right ) {\mathrm e}^{\frac {\left (c_{1} x +2 c_{0} \right ) x}{2}}}{2 c_{1}^{\frac {5}{2}}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (b_1 x+b_0) w_y + (c_1 y+c_0) w_z = a w + s_1 x + s_0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(b1*x+b0)*D[w[x,y,z],y]+(c1*y+c0)*D[w[x,y,z],z]==a*w[x,y,z]+s1*x+s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {a^2 \left (-e^{a x}\right ) c_1\left (-\text {b0} x-\frac {\text {b1} x^2}{2}+y,\frac {1}{2} \text {b0} \text {c1} x^2+\frac {1}{3} \text {b1} \text {c1} x^3-\text {c0} x-\text {c1} x y+z\right )+a \text {s0}+a \text {s1} x+\text {s1}}{a^2}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (b__1*x+b__0)*diff(w(x,y,z),y)+ (c__1*x+c__0)*diff(w(x,y,z),z)=a*w(x,y,z)+s__1*x+s__0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {a^{2} \mathit {\_F1} \left (-\frac {1}{2} b_{1} x^{2}-b_{0} x +y , -\frac {1}{2} c_{1} x^{2}-c_{0} x +z \right ) {\mathrm e}^{a x}+\left (-s_{1} x -s_{0} \right ) a -s_{1}}{a^{2}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a y+k_1 x+k_0) w_y + (b z+n_1 x+n_0) w_z = (c_1 x+c_0) w + s_1 x + s_0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(a*y+k1*x+k0)*D[w[x,y,z],y]+(b*z+n1*x+n0)*D[w[x,y,z],z]==(c1*x+c0)*w[x,y,z]+s1*x+s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (\frac {e^{-a x} \left (a^2 y+a (\text {k0}+\text {k1} x)+\text {k1}\right )}{a^2},\frac {e^{-b x} \left (b^2 z+b (\text {n0}+\text {n1} x)+\text {n1}\right )}{b^2}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a*y+k__1*x+k__0)*diff(w(x,y,z),y)+ (b*z+n__1*x+n__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x+s__0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (2 c_{1}^{\frac {5}{2}} \mathit {\_F1} \left (\frac {\left (a^{2} y +\left (k_{1} x +k_{0} \right ) a +k_{1} \right ) {\mathrm e}^{-a x}}{a^{2}}, \frac {\left (b^{2} z +\left (n_{1} x +n_{0} \right ) b +n_{1} \right ) {\mathrm e}^{-b x}}{b^{2}}\right )-2 c_{1}^{\frac {3}{2}} s_{1} \,{\mathrm e}^{-\frac {1}{2} c_{1} x^{2}-c_{0} x}+\sqrt {2}\, \sqrt {\pi }\, \left (-c_{0} s_{1} +c_{1} s_{0} \right ) c_{1} \erf \left (\frac {\sqrt {2}\, \left (\sqrt {c_{1}}\, x +\frac {c_{0}}{\sqrt {c_{1}}}\right )}{2}\right ) {\mathrm e}^{\frac {c_{0}^{2}}{2 c_{1}}}\right ) {\mathrm e}^{\frac {\left (c_{1} x +2 c_{0} \right ) x}{2}}}{2 c_{1}^{\frac {5}{2}}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_2 y+a_1 x+a_0) w_y + (b_3 z+b_2 y+b_1 x+b_0) w_z = (c_3 z+c_2 y+c_1 x+c_0) w + s_3 z + s_2 y+s_1 x+s_0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(a2*y+a1*x+a0)*D[w[x,y,z],y]+(b3*z+b2*y+b1*x+b0)*D[w[x,y,z],z]==(c3*z+c2*y+c1*x+c0)*w[x,y,z]+s3*z+s2*y+s1*x+s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
$Aborted
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__2*y+a__1*x+a__0)*diff(w(x,y,z),y)+ (b__3*z+b__2*y+b__1*x+b__0)*diff(w(x,y,z),z)=(c__3*z+c__2*y+c__1*x+c__0)*w(x,y,z)+s__3*z+s__2*y+s__1*x+s__0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
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Added Jan 6, 2020.
Problem Chapter 9.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x w_x + b x w_y + c z w_z = (\alpha x+\beta ) w + p x+q \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x,y,z],x]+b*x*D[w[x,y,z],y]+c*z*D[w[x,y,z],z]==(alpha*x+beta)*w[x,y,z]+p*x+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \operatorname {Gamma}\left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \operatorname {Gamma}\left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )+a \alpha x^{\frac {\beta }{a}} c_1\left (y-\frac {b x}{a},z x^{-\frac {c}{a}}\right )\right )}{a \alpha }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x*diff(w(x,y,z),x)+ b*x*diff(w(x,y,z),y)+ c*z*diff(w(x,y,z),z)=(alpha*x+beta)*w(x,y,z)+p*x+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = -\frac {3 \left (\left (a -\frac {\beta }{3}\right ) \left (\alpha x +a -\beta \right ) a^{2} q x^{-\frac {\beta }{a}-1} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{-\frac {a -\beta }{2 a}} \WhittakerM \left (\frac {-a -\beta }{2 a}, \frac {2 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}-\frac {4 \left (a -\frac {\beta }{2}\right )^{2} a \beta p x^{-\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{\frac {-2 a +\beta }{2 a}} \WhittakerM \left (\frac {2 a -\beta }{2 a}, \frac {3 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}}{3}+\left (a -\beta \right )^{2} \left (a -\frac {\beta }{3}\right ) a q x^{-\frac {\beta }{a}-1} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{-\frac {a -\beta }{2 a}} \WhittakerM \left (\frac {a -\beta }{2 a}, \frac {2 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}-2 \left (\frac {\left (\frac {\alpha x}{2}+a -\frac {\beta }{2}\right ) a^{2} p x^{-\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{\frac {-2 a +\beta }{2 a}} \WhittakerM \left (-\frac {\beta }{2 a}, \frac {3 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}}{3}+\left (a -\frac {\beta }{2}\right ) \left (a -\beta \right ) \left (a -\frac {\beta }{3}\right ) \alpha \mathit {\_F1} \left (\frac {a y -b x}{a}, z x^{-\frac {c}{a}}\right )\right ) \beta \right ) x^{\frac {\beta }{a}} {\mathrm e}^{\frac {\alpha x}{a}}}{\left (a -\beta \right ) \left (2 a -\beta \right ) \left (3 a -\beta \right ) \alpha \beta }\]
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Added Jan 6, 2020.
Problem Chapter 9.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x w_x + b y w_y + c z w_z = (\alpha x+\beta ) w + p x+q \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*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)*w[x,y,z]+p*x+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \operatorname {Gamma}\left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \operatorname {Gamma}\left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )+a \alpha x^{\frac {\beta }{a}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )\right )}{a \alpha }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x*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)*w(x,y,z)+p*x+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = -\frac {3 \left (\left (a -\frac {\beta }{3}\right ) \left (\alpha x +a -\beta \right ) a^{2} q x^{-\frac {\beta }{a}-1} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{-\frac {a -\beta }{2 a}} \WhittakerM \left (\frac {-a -\beta }{2 a}, \frac {2 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}-\frac {4 \left (a -\frac {\beta }{2}\right )^{2} a \beta p x^{-\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{\frac {-2 a +\beta }{2 a}} \WhittakerM \left (\frac {2 a -\beta }{2 a}, \frac {3 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}}{3}+\left (a -\beta \right )^{2} \left (a -\frac {\beta }{3}\right ) a q x^{-\frac {\beta }{a}-1} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{-\frac {a -\beta }{2 a}} \WhittakerM \left (\frac {a -\beta }{2 a}, \frac {2 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}-2 \left (\frac {\left (\frac {\alpha x}{2}+a -\frac {\beta }{2}\right ) a^{2} p x^{-\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{\frac {\beta }{a}} \left (\frac {\alpha }{a}\right )^{-\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{\frac {-2 a +\beta }{2 a}} \WhittakerM \left (-\frac {\beta }{2 a}, \frac {3 a -\beta }{2 a}, \frac {\alpha x}{a}\right ) {\mathrm e}^{-\frac {\alpha x}{2 a}}}{3}+\left (a -\frac {\beta }{2}\right ) \left (a -\beta \right ) \left (a -\frac {\beta }{3}\right ) \alpha \mathit {\_F1} \left (y x^{-\frac {b}{a}}, z x^{-\frac {c}{a}}\right )\right ) \beta \right ) x^{\frac {\beta }{a}} {\mathrm e}^{\frac {\alpha x}{a}}}{\left (a -\beta \right ) \left (2 a -\beta \right ) \left (3 a -\beta \right ) \alpha \beta }\]
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Added Jan 6, 2020.
Problem Chapter 9.2.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ x w_x + a z w_y + b y w_z = (c x+k) w + p x+q \]
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*x+k)*w[x,y,z]+p*x+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{c x} \left (-(c x)^k (p \operatorname {Gamma}(1-k,c x)+c q \operatorname {Gamma}(-k,c x))+c x^k c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right )}{c}\right \}\right \}\]
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*x+k)*w(x,y,z)+p*x+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int _{}^{y}\frac {\left (p x \left (\frac {\mathit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )^{\frac {1}{\sqrt {a b}}} \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}+q \right ) {\mathrm e}^{-\left (\int \frac {c x \left (\frac {\mathit {\_b} a b +\sqrt {a b}\, \sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}{\sqrt {a b}}\right )^{\frac {1}{\sqrt {a b}}} \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}+k}{\sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}d \mathit {\_b} \right )}}{\sqrt {\left (a z^{2}+\left (\mathit {\_b}^{2}-y^{2}\right ) b \right ) a}}d\mathit {\_b} +\mathit {\_F1} \left (\frac {a z^{2}-b y^{2}}{a}, x \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}\right )\right ) {\mathrm e}^{\int _{}^{y}\frac {c x \left (\frac {\mathit {\_a} a b +\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )^{\frac {1}{\sqrt {a b}}} \left (a z +\sqrt {a b}\, y \right )^{-\frac {\sqrt {a b}}{a b}}+k}{\sqrt {\left (a z^{2}+\left (\mathit {\_a}^{2}-y^{2}\right ) b \right ) a}}d\mathit {\_a}}\]
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