Added June 28, 2019.
Problem Chapter 8.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c w_z = (\lambda x^2 +\beta y^2+\gamma z^2+\delta ) w \]
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]== (alpha*x^2+beta*y^2+gamma*z^2+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right ) \exp \left (\frac {1}{3} \left (\frac {\alpha x^3+3 \delta x}{a}+\frac {\beta y^3}{b}+\frac {\gamma z^3}{c}\right )\right )\right \}\right \}\]
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^2+beta*y^2+gamma*z^2+delta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {z a -c x}{a}\right ) {\mathrm e}^{\frac {\left (\left (\frac {\alpha x^{2}}{3}+\beta y^{2}+\gamma z^{2}+\delta \right ) a^{2}-\left (b \beta y +c \gamma z \right ) a x +\frac {\left (b^{2} \beta +\gamma c^{2}\right ) x^{2}}{3}\right ) x}{a^{3}}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^2+a_0) w_y + (b_1 x^2+b_0) w_z = (\lambda x +\beta y+\gamma z+\delta ) w \]
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]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\text {a0} x-\frac {\text {a1} x^3}{3}+y,-\text {b0} x-\frac {\text {b1} x^3}{3}+z\right ) \exp \left (-\frac {1}{4} x \left (2 \text {a0} \beta x+\text {a1} \beta x^3-2 \alpha x+2 \text {b0} \gamma x+\text {b1} \gamma x^3-4 \beta y-4 \delta -4 \gamma z\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a1*x^2+a0)*diff(w(x,y,z),y)+(b1*x^2+b0)*diff(w(x,y,z),z)= (alpha*x+beta*y+gamma*z+delta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\frac {1}{3} \mathit {a1} x^{3}-\mathit {a0} x +y , -\frac {1}{3} \mathit {b1} x^{3}-\mathit {b0} x +z \right ) {\mathrm e}^{-\frac {\left (-2 \alpha x +\left (\mathit {a1} x^{3}+2 \mathit {a0} x -4 y \right ) \beta -4 \delta +\left (\mathit {b1} x^{3}+2 \mathit {b0} x -4 z \right ) \gamma \right ) x}{4}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y+ k_1 x^2+k_0) w_y + (b z+s_1 x^2+s_0) w_z = (c_1 x^2+c_0) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+k1*x^2+k0)*D[w[x, y,z], y] +(b*z+s1*x^2+s0)*D[w[x,y,z],z]== (c1*x^2+c0)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\text {c0} x+\frac {\text {c1} x^3}{3}} c_1\left (\frac {e^{-a x} \left (a^2 \left (\text {k0}+\text {k1} x^2\right )+a^3 y+2 a \text {k1} x+2 \text {k1}\right )}{a^3},\frac {e^{-b x} \left (b^2 \left (\text {s0}+\text {s1} x^2\right )+b^3 z+2 b \text {s1} x+2 \text {s1}\right )}{b^3}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a*y+k1*x^2+k0)*diff(w(x,y,z),y)+(b*z+s1*x^2+s0)*diff(w(x,y,z),z)= (c1*x^2+c0)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {\left (a^{3} y +2 a \mathit {k1} x +\left (\mathit {k1} x^{2}+\mathit {k0} \right ) a^{2}+2 \mathit {k1} \right ) {\mathrm e}^{-a x}}{a^{3}}, \frac {\left (b^{3} z +2 b \mathit {s1} x +\left (\mathit {s1} x^{2}+\mathit {s0} \right ) b^{2}+2 \mathit {s1} \right ) {\mathrm e}^{-b x}}{b^{3}}\right ) {\mathrm e}^{\frac {\left (x^{2} \mathit {c1} +3 \mathit {c0} \right ) x}{3}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_2 x y+ a_1 x^2+a_0) w_y + (b_2 x y+b_1 x^2+b_0) w_z = (c_2 y+c_1 z+c_0 x+s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*x*y+a1*x^2+a0)*D[w[x, y,z], y] +(b2*x*y+b1*x^2+b0)*D[w[x,y,z],z]== (c2*x+c1*z+c0*x+s)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\text {a0} \text {b2} x}{\text {a1}}-\frac {\text {b2} y}{\text {a1}}-\text {b0} x-\frac {\text {b1} x^3}{3}+\frac {\text {b2} x^3}{3}+z,e^{-\frac {\text {a1} x^2}{2}} (x+y)-\frac {\sqrt {\frac {\pi }{2}} (\text {a0}+1) \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\sqrt {\text {a1}}}\right ) \exp \left (\frac {2 (\text {a0}+1) \text {a1} \text {b2} \text {c1} x^2 \operatorname {HypergeometricPFQ}\left (\{1,1\},\left \{\frac {3}{2},2\right \},\frac {\text {a1} x^2}{2}\right )+2 \text {b2} \text {c1} e^{-\frac {\text {a1} x^2}{2}} \operatorname {Erfi}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right ) \left (\sqrt {2 \pi } \sqrt {\text {a1}} (x+y)-\pi (\text {a0}+1) e^{\frac {\text {a1} x^2}{2}} \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )\right )+\text {a1} x \left (2 \text {b2} \text {c1} ((\text {a0}-1) x-2 y)+\text {a1} \left (-2 \text {b0} \text {c1} x-\text {b1} \text {c1} x^3+\text {b2} \text {c1} x^3+2 \text {c0} x+4 \text {c1} z+2 \text {c2} x+4 s\right )\right )}{4 \text {a1}^2}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a1*x*y+a1*x^2+a0)*diff(w(x,y,z),y)+(b2*x*y+b1*x^2+b0)*diff(w(x,y,z),z)= (c2*x+c1*z+c0*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 ) = \mathit {\_F1} \left (\frac {\left (x +y \right ) \sqrt {\mathit {a1}}\, {\mathrm e}^{-\frac {\mathit {a1} x^{2}}{2}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\mathit {a0} +1\right ) \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )}{2}}{\sqrt {\mathit {a1}}}, -\frac {3 \left (-\frac {\sqrt {2}\, \left (\mathit {a0} +1\right ) \left (\frac {\sqrt {\pi }\, \sqrt {\frac {\mathit {a1}}{\pi }}}{\sqrt {\mathit {a1}}}-1\right ) \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )}{2}+\sqrt {\frac {\mathit {a1}}{\pi }}\, \left (x +y \right ) {\mathrm e}^{-\frac {\mathit {a1} x^{2}}{2}}\right ) \mathit {b2} \,{\mathrm e}^{\frac {\mathit {a1} x^{2}}{2}}+\left (-\left (\mathit {a1} x^{2}+3 \mathit {a0} +3\right ) \mathit {b2} x +\left (\mathit {b1} x^{3}+3 \mathit {b0} x -3 z \right ) \mathit {a1} \right ) \sqrt {\frac {\mathit {a1}}{\pi }}}{3 \sqrt {\frac {\mathit {a1}}{\pi }}\, \mathit {a1}}\right ) {\mathrm e}^{\int _{}^{x}\frac {-2 \left (\mathit {\_a}^{2} \mathit {a1} +3 \mathit {a0} +3\right ) \mathit {\_a} \mathit {b2} \mathit {c1} -3 \left (\left (-2 x -2 y \right ) {\mathrm e}^{-\frac {\mathit {a1} x^{2}}{2}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\mathit {a0} +1\right ) \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )}{\sqrt {\mathit {a1}}}\right ) \mathit {b2} \mathit {c1} \,{\mathrm e}^{\frac {\mathit {\_a}^{2} \mathit {a1}}{2}}+6 \left (\frac {\mathit {\_a}^{3} \mathit {b1} \mathit {c1}}{3}+\left (\mathit {b0} \mathit {c1} +\mathit {c0} +\mathit {c2} \right ) \mathit {\_a} +s \right ) \mathit {a1} -\frac {2 \left (-\frac {3 \sqrt {2}\, \left (\mathit {a0} +1\right ) \mathit {b2} \erf \left (\frac {\sqrt {2}\, \mathit {\_a} \sqrt {\mathit {a1}}}{2}\right ) {\mathrm e}^{\frac {\mathit {\_a}^{2} \mathit {a1}}{2}}}{2}-\frac {3 \sqrt {2}\, \left (\mathit {a0} +1\right ) \left (\frac {\sqrt {\pi }\, \sqrt {\frac {\mathit {a1}}{\pi }}}{\sqrt {\mathit {a1}}}-1\right ) \mathit {b2} \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right ) {\mathrm e}^{\frac {\mathit {a1} x^{2}}{2}}}{2}+3 \sqrt {\frac {\mathit {a1}}{\pi }}\, \left (x +y \right ) \mathit {b2} \,{\mathrm e}^{-\frac {\mathit {a1} x^{2}}{2}} {\mathrm e}^{\frac {\mathit {a1} x^{2}}{2}}+\left (-\left (\mathit {a1} x^{2}+3 \mathit {a0} +3\right ) \mathit {b2} x +\left (\mathit {b1} x^{3}+3 \mathit {b0} x -3 z \right ) \mathit {a1} \right ) \sqrt {\frac {\mathit {a1}}{\pi }}\right ) \mathit {c1}}{\sqrt {\frac {\mathit {a1}}{\pi }}}}{6 \mathit {a1}}d\mathit {\_a}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + c z w_z = x(\lambda x+\beta y+\gamma z) w \]
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]== x*(lambda*x+beta*y+gama*z)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right ) e^{\frac {\beta x y}{a+b}+\frac {\text {gama} x z}{a+c}+\frac {\lambda x^2}{2 a}}\right \}\right \}\]
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)= x*(lambda*x+beta*y+gama*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y x^{-\frac {b}{a}}, z x^{-\frac {c}{a}}\right ) {\mathrm e}^{\frac {\left (b c \lambda x +\left (2 \beta y +2 \mathit {gama} z +\lambda x \right ) a^{2}+\left (2 b \mathit {gama} z +2 \beta c y +\left (b +c \right ) \lambda x \right ) a \right ) x}{2 \left (a +b \right ) \left (a +c \right ) a}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2 w_x + b x y w_y + c x z w_z = (\lambda x+\beta y+\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*D[w[x, y,z], x] + b*x*y*D[w[x, y,z], y] +c*x*z*D[w[x,y,z],z]== (lambda*x+beta*y+gama*z)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x^{\frac {\lambda }{a}} e^{-\frac {\frac {\beta y}{a-b}+\frac {\text {gama} z}{a-c}}{x}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x^2*diff(w(x,y,z),x)+b*x*y*diff(w(x,y,z),y)+c*x*z*diff(w(x,y,z),z)= (lambda*x+beta*y+gama*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = x^{\frac {\lambda }{a}} \mathit {\_F1} \left (y x^{-\frac {b}{a}}, z x^{-\frac {c}{a}}\right ) {\mathrm e}^{\frac {-\left (a -c \right ) \beta y -\left (a -b \right ) \mathit {gama} z}{\left (a -b \right ) \left (a -c \right ) x}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2 w_x + b x y w_y + c z^2 w_z = k y^2 w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*D[w[x, y,z], x] + b*x*y*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== k*y^2*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{-\frac {k y^2}{a x-2 b x}} c_1\left (y x^{-\frac {b}{a}},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x^2*diff(w(x,y,z),x)+b*x*y*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)= k*y^2*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y x^{-\frac {b}{a}}, \frac {a x -c z}{a x z}\right ) {\mathrm e}^{-\frac {k y^{2}}{\left (a -2 b \right ) x}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2 w_x + b y^2 w_y + c z^2 w_z = k x y w \]
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*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \left (\frac {a x}{y}\right )^{\frac {k x y}{a x-b y}} c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \}\]
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*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\frac {a x}{y}\right )^{\frac {k x y}{a x -b y}} \mathit {\_F1} \left (\frac {a x -b y}{a x y}, \frac {a x -c z}{a x z}\right )\]
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Added June 28, 2019.
Problem Chapter 8.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x^2 w_x + b y^2 w_y + c z^2 w_z = (\lambda x^2+\beta y^2 + \gamma z^2) w \]
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]== (lambda*x^2+beta*y^2+gamma*z^2)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right ) \exp \left (\frac {\beta y^2}{b y-a x}+\frac {\gamma z^2}{c z-a x}+\frac {\lambda x}{a}\right )\right \}\right \}\]
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)= (lambda*x^2+beta*y^2+gamma*z^2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a x -b y}{a x y}, \frac {a x -c z}{a x z}\right ) {\mathrm e}^{\frac {b c \lambda x y z +\left (-\beta y^{2}-\gamma z^{2}+\lambda x^{2}\right ) a^{2} x +\left (\beta c y^{2} z -c \lambda x^{2} z -\left (-\gamma z^{2}+\lambda x^{2}\right ) b y \right ) a}{\left (a x -c z \right ) \left (a x -b y \right ) a}}\]
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