Added Jan 6, 2020.
Problem Chapter 9.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^2+a_0) w_y + (b_1 x^2+b_0)w_z = (c_1 x+c_0) w + s_1 x^2+s_0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(a1*x^2+a0)*D[w[x,y,z],y]+(b1*x^2+b0)*D[w[x,y,z],z]==(c1*x+c0)*w[x,y,z]+s1*x^2+s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (-\text {a0} x-\frac {\text {a1} x^3}{3}+y,-\text {b0} x-\frac {\text {b1} x^3}{3}+z\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) \left (\text {c0}^2 \text {s1}+\text {c1}^2 \text {s0}+\text {c1} \text {s1}\right )}{\text {c1}^{5/2}}+\frac {\text {s1} (\text {c0}-\text {c1} x)}{\text {c1}^2}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*x^2+a__0)*diff(w(x,y,z),y)+ (b__1*x^2+b__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x^2+s_0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (c_{1}^{\frac {9}{2}} \mathit {\_F1} \left (-\frac {1}{3} a_{1} x^{3}-a_{0} x +y , -\frac {1}{3} b_{1} x^{3}-b_{0} x +z \right )+\frac {\sqrt {\pi }\, \sqrt {2}\, \left (c_{0}^{2} s_{1} +c_{1}^{2} \mathit {s\_0} +c_{1} s_{1} \right ) c_{1}^{2} \erf \left (\frac {\sqrt {2}\, \left (\sqrt {c_{1}}\, x +\frac {c_{0}}{\sqrt {c_{1}}}\right )}{2}\right ) {\mathrm e}^{\frac {c_{0}^{2}}{2 c_{1}}}}{2}+\left (-c_{1}^{\frac {7}{2}} x +c_{0} c_{1}^{\frac {5}{2}}\right ) s_{1} \,{\mathrm e}^{-\frac {1}{2} c_{1} x^{2}-c_{0} x}\right ) {\mathrm e}^{\frac {\left (c_{1} x +2 c_{0} \right ) x}{2}}}{c_{1}^{\frac {9}{2}}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (b_1 x^2+b_0) w_y + (c_1 y^2+c_0)w_z = a w + s_1 x^2+s_0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(b1*x^2+b0)*D[w[x,y,z],y]+(c1*y^2+c0)*D[w[x,y,z],z]==a*w[x,y,z]+s1*x^2+s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {a^3 e^{a x} c_1\left (-\text {b0} x-\frac {\text {b1} x^3}{3}+y,-\frac {1}{3} \text {b0}^2 \text {c1} x^3-\frac {3}{10} \text {b0} \text {b1} \text {c1} x^5+\text {b0} \text {c1} x^2 y-\frac {1}{14} \text {b1}^2 \text {c1} x^7+\frac {1}{2} \text {b1} \text {c1} x^4 y-\text {c0} x-\text {c1} x y^2+z\right )-a^2 \left (\text {s0}+\text {s1} x^2\right )-2 a \text {s1} x-2 \text {s1}}{a^3}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (b__1*x^2+b__0)*diff(w(x,y,z),y)+ (c__1*y^2+c__0)*diff(w(x,y,z),z)=a*w(x,y,z)+s__1*x^2+s_0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {a^{3} \mathit {\_F1} \left (-\frac {1}{3} b_{1} x^{3}-b_{0} x +y , -c_{0} x -\frac {\left (b_{1}^{2} x^{6}+\frac {21}{5} b_{0} b_{1} x^{4}-7 b_{1} x^{3} y +\frac {14}{3} b_{0}^{2} x^{2}-14 b_{0} x y +14 y^{2}\right ) c_{1} x}{14}+z \right ) {\mathrm e}^{a x}-2 a s_{1} x +\left (-s_{1} x^{2}-\mathit {s\_0} \right ) a^{2}-2 s_{1}}{a^{3}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a y+k_1 x^2+k_0) w_y + (b z+n_1 x^2+n_0)w_z = (c_1 x+c_0) w + s_1 x+s_0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(a*y+k1*x^2+k0)*D[w[x,y,z],y]+(b*z+n1*x^2+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]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (\frac {e^{-a x} \left (a^2 \left (\text {k0}+\text {k1} x^2\right )+a^3 y+2 a \text {k1} x+2 \text {k1}\right )}{a^3},\frac {e^{-b x} \left (b^2 \left (\text {n0}+\text {n1} x^2\right )+b^3 z+2 b \text {n1} x+2 \text {n1}\right )}{b^3}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a*y+k__1*x^2+k__0)*diff(w(x,y,z),y)+ (b*z+n__1*x^2+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'));
\[w \left (x , y , z\right ) = -\frac {\left (-2 c_{1}^{\frac {5}{2}} \mathit {\_F1} \left (\frac {\left (a^{3} y +2 a k_{1} x +\left (k_{1} x^{2}+k_{0} \right ) a^{2}+2 k_{1} \right ) {\mathrm e}^{-a x}}{a^{3}}, \frac {\left (b^{3} z +2 b n_{1} x +\left (n_{1} x^{2}+n_{0} \right ) b^{2}+2 n_{1} \right ) {\mathrm e}^{-b x}}{b^{3}}\right )+2 c_{1}^{\frac {3}{2}} s_{1} \,{\mathrm e}^{-\frac {1}{2} c_{1} x^{2}-c_{0} x}+\sqrt {2}\, \sqrt {\pi }\, \left (c_{0} s_{1} -c_{1} \mathit {s\_0} \right ) c_{1} \erf \left (\frac {\sqrt {2}\, \left (\sqrt {c_{1}}\, x +\frac {c_{0}}{\sqrt {c_{1}}}\right )}{2}\right ) {\mathrm e}^{\frac {c_{0}^{2}}{2 c_{1}}}\right ) {\mathrm e}^{\frac {\left (c_{1} x +2 c_{0} \right ) x}{2}}}{2 c_{1}^{\frac {5}{2}}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_2 x y+a_1 x+a_0) w_y + (b_3 y z+b_2 y^2+b_1 x^2+b_0)w_z = (c_3 z+c_2 y+c_1 x+c_0) w + s_1 x y+s_2 x z \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+(a2*x*y+a1*x+a0)*D[w[x,y,z],y]+(b3*y*z+b2*y^2+b1*x^2+b0)*D[w[x,y,z],z]==(c3*z+c2*y+c1*x+c0)*w[x,y,z]+s1*x*y+s2*x*z; 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*x*y+a__1*x+a__0)*diff(w(x,y,z),y)+ (b__3*y*z+b__2*y^2+b__1*x^2+b__0)*diff(w(x,y,z),z)=(c__3*z+c__2*y+c__1*x+c__0)*w(x,y,z)+s__1*x*y+s__2*x*z; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
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Added Jan 6, 2020.
Problem Chapter 9.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x w_x + b x w_y + c z w_z = k x w + s x^2 \]
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]==k*x*w[x,y,z]+s*x^2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {s (a+k x)}{k^2}+e^{\frac {k x}{a}} c_1\left (y-\frac {b x}{a},z x^{-\frac {c}{a}}\right )\right \}\right \}\]
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)=k*x*w(x,y,z)+s*x^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {k^{2} \mathit {\_F1} \left (\frac {a y -b x}{a}, z x^{-\frac {c}{a}}\right ) {\mathrm e}^{\frac {k x}{a}}-\left (x k +a \right ) s}{k^{2}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x w_x + b y w_y + c z w_z = k x w + s x^2 \]
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]==k*x*w[x,y,z]+s*x^2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {s (a+k x)}{k^2}+e^{\frac {k x}{a}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )\right \}\right \}\]
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)=k*x*w(x,y,z)+s*x^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {k^{2} \mathit {\_F1} \left (y x^{-\frac {b}{a}}, z x^{-\frac {c}{a}}\right ) {\mathrm e}^{\frac {k x}{a}}-\left (x k +a \right ) s}{k^{2}}\]
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Added Jan 6, 2020.
Problem Chapter 9.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x^2 w_x + b y^2 w_y + c z^2 w_z = (k x+s) w + p x+q \]
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+s)*w[x,y,z]+p*x+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{-\frac {s}{a x}} \left (-\frac {s}{a x}\right )^{-\frac {k}{a}} \left (a s x^{\frac {k}{a}} \left (-\frac {s}{a x}\right )^{\frac {k}{a}} c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right )+p s \operatorname {Gamma}\left (\frac {k}{a},-\frac {s}{a x}\right )-a q \operatorname {Gamma}\left (\frac {a+k}{a},-\frac {s}{a x}\right )\right )}{a s}\right \}\right \}\]
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+s)*w(x,y,z)+p*x+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int \frac {\left (p x +q \right ) x^{\frac {-2 a -k}{a}} {\mathrm e}^{\frac {s}{a x}}}{a}d x +\mathit {\_F1} \left (\frac {a x -b y}{a x y}, \frac {a x -c z}{a x z}\right )\right ) x^{\frac {k}{a}} {\mathrm e}^{-\frac {s}{a x}}\]
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