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'));
\[w \left ( x,y,z \right ) ={\frac {1}{a\alpha }{{\rm e}^{{\frac {x \left ( \alpha \,x+2\,\beta \right ) }{2\,a}}}} \left ( -{\frac {\sqrt {\pi }\sqrt {2} \left ( -\alpha \,q+\beta \,p \right ) }{2}{{\rm e}^{{\frac {{\beta }^{2}}{2\,a\alpha }}}}\erf \left ( {\frac {\sqrt {2}x}{2}\sqrt {{\frac {\alpha }{a}}}}+{\frac {\beta \,\sqrt {2}}{2\,a}{\frac {1}{\sqrt {{\frac {\alpha }{a}}}}}} \right ) }+a\sqrt {{\frac {\alpha }{a}}} \left ( \alpha \,{\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {za-xc}{a}} \right ) -{{\rm e}^{-{\frac {x \left ( \alpha \,x+2\,\beta \right ) }{2\,a}}}}p \right ) \right ) {\frac {1}{\sqrt {{\frac {\alpha }{a}}}}}}\]
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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'));
\[w \left ( x,y,z \right ) = \left ( \int ^{y}\!-{ \left ( \left ( -px-q \right ) \sqrt {ab}+ \left ( \ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab} \right ) {\frac {1}{\sqrt {ab}}}} \right ) -\ln \left ( { \left ( ab{\it \_b}+\sqrt {ab}\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) } \right ) {\frac {1}{\sqrt {ab}}}} \right ) \right ) p \right ) {{\rm e}^{-{\int \!{ \left ( \sqrt {ab} \left ( xc+k \right ) +c \left ( \ln \left ( { \left ( ab{\it \_b}+\sqrt {ab}\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) } \right ) {\frac {1}{\sqrt {ab}}}} \right ) -\ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab} \right ) {\frac {1}{\sqrt {ab}}}} \right ) \right ) \right ) {\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }}}}\,{\rm d}{\it \_b}{\frac {1}{\sqrt {ab}}}}}}{\frac {1}{\sqrt {ab}}}{\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {a{z}^{2}-{y}^{2}b}{a}},-{ \left ( -x\sqrt {ab}+\ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab} \right ) {\frac {1}{\sqrt {ab}}}} \right ) \right ) {\frac {1}{\sqrt {ab}}}} \right ) \right ) {{\rm e}^{\int ^{y}\!{ \left ( \sqrt {ab} \left ( xc+k \right ) +c \left ( \ln \left ( { \left ( ab{\it \_a}+\sqrt {ab}\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) } \right ) {\frac {1}{\sqrt {ab}}}} \right ) -\ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ab} \right ) {\frac {1}{\sqrt {ab}}}} \right ) \right ) \right ) {\frac {1}{\sqrt {ab}}}{\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) }}}}{d{\it \_a}}}}\]
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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'));
\[w \left ( x,y,z \right ) ={\frac {1}{2} \left ( 2\,{\it \_F1} \left ( -1/2\,a_{1}\,{x}^{2}-a_{0}\,x+y,-1/2\,b_{1}\,{x}^{2}-b_{0}\,x+z \right ) {c_{1}}^{5/2}+c_{1}\,\sqrt {2}\sqrt {\pi }{{\rm e}^{{\frac {{c_{0}}^{2}}{2\,c_{1}}}}} \left ( -c_{0}\,s_{1}+c_{1}\,s_{0} \right ) \erf \left ( {\frac {\sqrt {2}}{2} \left ( \sqrt {c_{1}}x+{c_{0}{\frac {1}{\sqrt {c_{1}}}}} \right ) } \right ) -2\,{c_{1}}^{3/2}{{\rm e}^{-1/2\,c_{1}\,{x}^{2}-c_{0}\,x}}s_{1} \right ) {{\rm e}^{{\frac {x \left ( c_{1}\,x+2\,c_{0} \right ) }{2}}}}{c_{1}}^{-{\frac {5}{2}}}}\]
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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'));
\[w \left ( x,y,z \right ) ={\frac {1}{{a}^{2}} \left ( {{\rm e}^{ax}}{\it \_F1} \left ( -{\frac {b_{1}\,{x}^{2}}{2}}-b_{0}\,x+y,-{\frac {c_{1}\,{x}^{2}}{2}}-c_{0}\,x+z \right ) {a}^{2}+ \left ( -s_{1}\,x-s_{0} \right ) a-s_{1} \right ) }\]
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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'));
\[w \left ( x,y,z \right ) ={\frac {1}{2} \left ( 2\,{\it \_F1} \left ( {\frac {{{\rm e}^{-ax}} \left ( y{a}^{2}+a \left ( k_{1}\,x+k_{0} \right ) +k_{1} \right ) }{{a}^{2}}},{\frac {{{\rm e}^{-bx}} \left ( z{b}^{2}+b \left ( n_{1}\,x+n_{0} \right ) +n_{1} \right ) }{{b}^{2}}} \right ) {c_{1}}^{5/2}+c_{1}\,\sqrt {2}\sqrt {\pi }{{\rm e}^{{\frac {{c_{0}}^{2}}{2\,c_{1}}}}} \left ( -c_{0}\,s_{1}+c_{1}\,s_{0} \right ) \erf \left ( {\frac {\sqrt {2}}{2} \left ( \sqrt {c_{1}}x+{c_{0}{\frac {1}{\sqrt {c_{1}}}}} \right ) } \right ) -2\,{c_{1}}^{3/2}{{\rm e}^{-1/2\,c_{1}\,{x}^{2}-c_{0}\,x}}s_{1} \right ) {{\rm e}^{{\frac {x \left ( c_{1}\,x+2\,c_{0} \right ) }{2}}}}{c_{1}}^{-{\frac {5}{2}}}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \text {Gamma}\left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \text {Gamma}\left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )+a \alpha x^{\frac {\beta }{a}} c_1\left (y-\frac {b x}{a},z x^{-\frac {c}{a}}\right )\right )}{a \alpha }\right \}\right \}\]
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 ) =-3\,{\frac {1}{\alpha \, \left ( a-\beta \right ) \left ( 2\,a-\beta \right ) \left ( 3\,a-\beta \right ) \beta } \left ( {a}^{2} \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}} \left ( \alpha \,x+a-\beta \right ) \left ( a-\beta /3 \right ) \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}q \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {-a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -4/3\,{x}^{-{\frac {\beta }{a}}}a\beta \, \left ( a-\beta /2 \right ) ^{2} \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}}p \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {2\,a-\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) +a \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}} \left ( a-\beta \right ) ^{2} \left ( a-\beta /3 \right ) \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}q \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -2\, \left ( 1/3\,{x}^{-{\frac {\beta }{a}}}{a}^{2} \left ( 1/2\,\alpha \,x+a-\beta /2 \right ) \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}}p \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( -1/2\,{\frac {\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) + \left ( a-\beta \right ) \left ( a-\beta /3 \right ) \left ( a-\beta /2 \right ) {\it \_F1} \left ( {\frac {ya-bx}{a}},z{x}^{-{\frac {c}{a}}} \right ) \alpha \right ) \beta \right ) {{\rm e}^{{\frac {\alpha \,x}{a}}}}{x}^{{\frac {\beta }{a}}}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \text {Gamma}\left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \text {Gamma}\left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )+a \alpha x^{\frac {\beta }{a}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )\right )}{a \alpha }\right \}\right \}\]
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 ) =-3\,{\frac {1}{\alpha \, \left ( a-\beta \right ) \left ( 2\,a-\beta \right ) \left ( 3\,a-\beta \right ) \beta } \left ( {a}^{2} \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}} \left ( \alpha \,x+a-\beta \right ) \left ( a-\beta /3 \right ) \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}q \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {-a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -4/3\,{x}^{-{\frac {\beta }{a}}}a\beta \, \left ( a-\beta /2 \right ) ^{2} \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}}p \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {2\,a-\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) +a \left ( {\frac {\alpha \,x}{a}} \right ) ^{-1/2\,{\frac {a-\beta }{a}}} \left ( a-\beta \right ) ^{2} \left ( a-\beta /3 \right ) \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{x}^{-1-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}}q \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( 1/2\,{\frac {a-\beta }{a}},1/2\,{\frac {2\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) -2\,\beta \, \left ( 1/3\,{x}^{-{\frac {\beta }{a}}}{a}^{2} \left ( 1/2\,\alpha \,x+a-\beta /2 \right ) \left ( {\frac {\alpha }{a}} \right ) ^{-{\frac {\beta }{a}}}{{\rm e}^{-1/2\,{\frac {\alpha \,x}{a}}}} \left ( {\frac {\alpha \,x}{a}} \right ) ^{1/2\,{\frac {-2\,a+\beta }{a}}}p \left ( {\frac {\alpha }{a}} \right ) ^{{\frac {\beta }{a}}} \WhittakerM \left ( -1/2\,{\frac {\beta }{a}},1/2\,{\frac {3\,a-\beta }{a}},{\frac {\alpha \,x}{a}} \right ) + \left ( a-\beta \right ) \left ( a-\beta /3 \right ) \left ( a-\beta /2 \right ) \alpha \,{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) \right ) \right ) {x}^{{\frac {\beta }{a}}}{{\rm e}^{{\frac {\alpha \,x}{a}}}}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{c x} \left (-(c x)^k (p \text {Gamma}(1-k,c x)+c q \text {Gamma}(-k,c x))+c x^k c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right )}{c}\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) = \left ( \int ^{y}\!{{{\rm e}^{-\int \!{ \left ( x \left ( \sqrt {ab}y+az \right ) ^{-{\frac {1}{ab}\sqrt {ab}}} \left ( { \left ( ab{\it \_b}+\sqrt {ab}\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) } \right ) {\frac {1}{\sqrt {ab}}}} \right ) ^{{\frac {1}{\sqrt {ab}}}}c+k \right ) {\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }}}}\,{\rm d}{\it \_b}}} \left ( px \left ( \sqrt {ab}y+az \right ) ^{-{\frac {1}{ab}\sqrt {ab}}} \left ( { \left ( ab{\it \_b}+\sqrt {ab}\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) } \right ) {\frac {1}{\sqrt {ab}}}} \right ) ^{{\frac {1}{\sqrt {ab}}}}+q \right ) {\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_b}}^{2}-{y}^{2} \right ) b \right ) }}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {a{z}^{2}-{y}^{2}b}{a}},x \left ( \sqrt {ab}y+az \right ) ^{-{\frac {1}{ab}\sqrt {ab}}} \right ) \right ) {{\rm e}^{\int ^{y}\!{ \left ( x \left ( \sqrt {ab}y+az \right ) ^{-{\frac {1}{ab}\sqrt {ab}}} \left ( { \left ( ab{\it \_a}+\sqrt {ab}\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) } \right ) {\frac {1}{\sqrt {ab}}}} \right ) ^{{\frac {1}{\sqrt {ab}}}}c+k \right ) {\frac {1}{\sqrt {a \left ( a{z}^{2}+ \left ( {{\it \_a}}^{2}-{y}^{2} \right ) b \right ) }}}}{d{\it \_a}}}}\]
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