Added June 27, 2019.
Problem Chapter 8.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c w_z = (\alpha x+\beta y+\gamma z+ \delta ) w \]
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+beta*y+gamma*z+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right ) \exp \left (\frac {x (a (\alpha x+2 \beta y+2 \delta +2 \gamma z)-x (b \beta +c \gamma ))}{2 a^2}\right )\right \}\right \}\]
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+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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a z -c x}{a}\right ) {\mathrm e}^{\frac {\left (\left (\frac {\alpha x}{2}+\beta y +\gamma z +\delta \right ) a -\frac {\left (b \beta +c \gamma \right ) x}{2}\right ) x}{a^{2}}}\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a z w_y + b y w_z = (c x+s) w \]
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+s)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^2}{2}+s x} c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )\right \}\right \}\]
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+s)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \,z^{2}-b \,y^{2}}{a}, -\frac {-\sqrt {a b}\, x +\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (\ln \left (\frac {\textit {\_a} a b +\sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )\right ) c +\sqrt {a b}\, \left (c x +s \right )}{\sqrt {a b}\, \sqrt {\left (a \,z^{2}+\left (\textit {\_a}^{2}-y^{2}\right ) b \right ) a}}d \textit {\_a}}\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x+a_0) w_y + (b_1 x+b_0) w_z = (\alpha x+\beta y+\gamma z+\delta ) w \]
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]== (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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\text {a0} x-\frac {\text {a1} x^2}{2}+y,-\text {b0} x-\frac {\text {b1} x^2}{2}+z\right ) \exp \left (\frac {1}{6} x \left (-3 \text {a0} \beta x-2 \text {a1} \beta x^2+3 \alpha x-3 \text {b0} \gamma x-2 \text {b1} \gamma x^2+6 \beta y+6 \delta +6 \gamma z\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a1*x+a0)*diff(w(x,y,z),y)+ (b1*x+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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (-\frac {1}{2} \mathit {a1} \,x^{2}-\mathit {a0} x +y , -\frac {1}{2} \mathit {b1} \,x^{2}-\mathit {b0} x +z \right ) {\mathrm e}^{-\frac {\left (-\frac {3 \alpha x}{2}+\left (\mathit {a1} \,x^{2}+\frac {3}{2} \mathit {a0} x -3 y \right ) \beta -3 \delta +\left (\mathit {b1} \,x^{2}+\frac {3}{2} \mathit {b0} x -3 z \right ) \gamma \right ) x}{3}}\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_2 y+a_1 x+a_0) w_y + (b_2 y+b_1 x+b_0) w_z = (c_2 y+c_1 z+c_0 x+s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a2*y+a1*x+a0)*D[w[x, y,z], y] + (b2*y+b1*x+b0)*D[w[x,y,z],z]== (c2*y+c1*z+c0*x+s)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\text {a2} x} (\text {a2} (\text {a0}+\text {a2} y)+\text {a1} \text {a2} x+\text {a1})}{\text {a2}^2},\frac {e^{-\text {a2} x} \left (\text {a2} \left (2 \text {a0} \text {b2} \left (\text {a2} x e^{\text {a2} x}+1\right )-\text {a2} \left (\text {a2} e^{\text {a2} x} \left (2 \text {b0} x+\text {b1} x^2-2 z\right )+2 \text {b2} y \left (e^{\text {a2} x}-1\right )\right )\right )+\text {a1} \text {b2} \left (\text {a2}^2 x^2 e^{\text {a2} x}+2 \text {a2} x+2\right )\right )}{2 \text {a2}^3}\right ) \exp \left (\frac {\text {a2} \left (3 \text {a0} \text {b2} \text {c1} \left (\text {a2}^2 x^2-2 \text {a2} x+2\right )-6 \text {a0} \text {a2} \text {c2} (\text {a2} x-1)+\text {a2}^3 x \left (-3 \text {b0} \text {c1} x-2 \text {b1} \text {c1} x^2+3 \text {c0} x+6 \text {c1} z+6 s\right )+6 \text {a2}^2 y (\text {c2}-\text {b2} \text {c1} x)+6 \text {a2} \text {b2} \text {c1} y\right )+\text {a1} \left (\text {b2} \text {c1} \left (2 \text {a2}^3 x^3-3 \text {a2}^2 x^2+6\right )-3 \text {a2} \text {c2} \left (\text {a2}^2 x^2-2\right )\right )}{6 \text {a2}^4}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a2*y+a1*x+a0)*diff(w(x,y,z),y)+ (b2*y+b1*x+b0)*diff(w(x,y,z),z)= (c2*y+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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\left (\mathit {a2}^{2} y +\mathit {a1} +\left (\mathit {a1} x +\mathit {a0} \right ) \mathit {a2} \right ) {\mathrm e}^{-\mathit {a2} x}}{\mathit {a2}^{2}}, \frac {-2 \mathit {a0} \mathit {a2} \mathit {b2} +\left (-\mathit {b1} \,x^{2}-2 \mathit {b0} x +2 z \right ) \mathit {a2}^{3}+\left (\mathit {a1} \,x^{2}+2 \mathit {a0} x -2 y \right ) \mathit {a2}^{2} \mathit {b2} -2 \mathit {a1} \mathit {b2}}{2 \mathit {a2}^{3}}\right ) {\mathrm e}^{\frac {6 \left (\frac {\left (-\mathit {a1} \mathit {c2} +\mathit {a2} \mathit {c0} \right ) \mathit {a2}^{2} x^{2}}{2}+\left (\mathit {a2}^{2} y +\mathit {a0} \mathit {a2} +\mathit {a1} \right ) \mathit {c2} +\left (-\mathit {a0} \,\mathit {a2}^{2} \mathit {c2} +\mathit {a2}^{3} s \right ) x \right ) \mathit {a2} +\left (-2 \left (\mathit {b1} \,x^{2}+\frac {3}{2} \mathit {b0} x -3 z \right ) \mathit {a2}^{4} x +\left (2 \mathit {a1} \,\mathit {a2}^{3} x^{3}+6 \mathit {a2}^{2} y +6 \mathit {a0} \mathit {a2} +\left (3 \mathit {a0} \,\mathit {a2}^{3}-3 \mathit {a1} \,\mathit {a2}^{2}\right ) x^{2}+6 \mathit {a1} +\left (-6 \mathit {a2}^{3} y -6 \mathit {a0} \,\mathit {a2}^{2}\right ) x \right ) \mathit {b2} \right ) \mathit {c1}}{6 \mathit {a2}^{4}}}\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y+k_1 x+k_0) w_y + (b z+s_1 x+s_0) w_z = (c_1 x+c_0) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+k1*x+k0)*D[w[x, y,z], y] + (b*z+s1*x+s0)*D[w[x,y,z],z]== (c1*x+c0)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\text {c0} x+\frac {\text {c1} x^2}{2}} c_1\left (\frac {e^{-a x} \left (a^2 y+a (\text {k0}+\text {k1} x)+\text {k1}\right )}{a^2},\frac {e^{-b x} \left (b^2 z+b (\text {s0}+\text {s1} x)+\text {s1}\right )}{b^2}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a*y+k1*x+k0)*diff(w(x,y,z),y)+(b*z+s1*x+s0)*diff(w(x,y,z),z)= (c1*x+c0)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\left (a^{2} y +\left (\mathit {k1} x +\mathit {k0} \right ) a +\mathit {k1} \right ) {\mathrm e}^{-a x}}{a^{2}}, \frac {\left (b^{2} z +\left (\mathit {s1} x +\mathit {s0} \right ) b +\mathit {s1} \right ) {\mathrm e}^{-b x}}{b^{2}}\right ) {\mathrm e}^{\frac {\left (\mathit {c1} x +2 \mathit {c0} \right ) x}{2}}\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + c z w_z = (\alpha x+\beta y+\gamma z+\delta ) w \]
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]== (alpha*x+beta*y+gamma*z+beta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x^{\frac {\beta }{a}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right ) e^{\frac {\alpha x}{a}+\frac {\beta y}{b}+\frac {\gamma z}{c}}\right \}\right \}\]
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*y+gamma*z+beta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = x^{\frac {\beta }{a}} \textit {\_F1} \left (y \,x^{-\frac {b}{a}}, z \,x^{-\frac {c}{a}}\right ) {\mathrm e}^{\frac {a b \gamma z +\left (a \beta y +\alpha b x \right ) c}{a b c}}\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + a z w_y + b y w_z = c w \]
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*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x^c c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := x*diff(w(x,y,z),x)+ a*z*y*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= c*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \,z^{2}-2 b y}{a}, x \,{\mathrm e}^{\frac {2 \arctanh \left (\frac {z}{\sqrt {\frac {a \,z^{2}-2 b y}{a}}}\right )}{\sqrt {\frac {a \,z^{2}-2 b y}{a}}\, a}}\right ) {\mathrm e}^{-\frac {2 c \arctanh \left (\frac {\sqrt {a^{2} z^{2}}}{\sqrt {\frac {a \,z^{2}-2 b y}{a}}\, a}\right )}{\sqrt {\frac {a \,z^{2}-2 b y}{a}}\, a}}\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a b x w_x + b(a y + b z) w_y + a(a y -b z) w_z = c w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*b*x*D[w[x, y,z], x] + b*(a*y+b*z)*D[w[x, y,z], y] + a*(a*y-b*z)*D[w[x,y,z],z]== c*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x^{\frac {c}{a b}} c_1\left (-\frac {x^{\sqrt {2}} \left (\left (\sqrt {2}-2\right ) a y+\sqrt {2} b z\right )}{4 a},\frac {x^{-\sqrt {2}} \left (\left (2+\sqrt {2}\right ) a y+\sqrt {2} b z\right )}{4 a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*b*x*diff(w(x,y,z),x)+ b*(a*y+b*z)*diff(w(x,y,z),y)+a*(a*y-b*z)*diff(w(x,y,z),z)= c*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\frac {\sqrt {2}\, a^{2} y}{\left (-a^{2} y^{2}+2 a b y z +b^{2} z^{2}\right ) \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}+\frac {a y}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}+\frac {b z}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}\right )^{\frac {\sqrt {2}\, c}{2 \sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}\, b}} \textit {\_F1} \left (-\frac {1}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}, x \left (\frac {\frac {\sqrt {2}\, a^{2} y}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}+\left (\frac {a y}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}+\frac {b z}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}\right ) \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}{\sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}\right )^{-\frac {\sqrt {2}\, a}{2 \sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}\, \sqrt {\frac {a^{2}}{-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}}}\right )\]
____________________________________________________________________________________
Added June 27, 2019.
Problem Chapter 8.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (a_1 x +a_0) w_x + (b_1 y + b_0) w_y + (c_1 z +c_0) w_z = (\alpha x+\beta y+\gamma z+\delta ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a1*x+a0)*D[w[x, y,z], x] + (b1*y+b0)*D[w[x, y,z], y] +(c1*z+c0)*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]];
\[\left \{\left \{w(x,y,z)\to (\text {a0}+\text {a1} x)^{-\frac {\text {a0} \alpha \text {b1} \text {c1}+\text {a1} \text {b0} \beta \text {c1}+\text {a1} \text {b1} \text {c0} \gamma -\text {a1} \text {b1} \text {c1} \delta }{\text {a1}^2 \text {b1} \text {c1}}} c_1\left (\frac {(\text {b0}+\text {b1} y) (\text {a0}+\text {a1} x)^{-\frac {\text {b1}}{\text {a1}}}}{\text {b1}},\frac {(\text {c0}+\text {c1} z) (\text {a0}+\text {a1} x)^{-\frac {\text {c1}}{\text {a1}}}}{\text {c1}}\right ) \exp \left (\frac {\alpha x}{\text {a1}}+\frac {\beta (\text {b0}+\text {b1} y)}{\text {b1}^2}+\frac {\gamma (\text {c0}+\text {c1} z)}{\text {c1}^2}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := (a1*x+a0)*diff(w(x,y,z),x)+(b1*y+b0)*diff(w(x,y,z),y)+(c1*z+c0)*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'));
\[w \left (x , y , z\right ) = \left (\mathit {a1} x +\mathit {a0} \right )^{-\frac {\mathit {a0} \alpha }{\mathit {a1}^{2}}-\frac {\mathit {b0} \beta }{\mathit {a1} \mathit {b1}}-\frac {\mathit {c0} \gamma }{\mathit {a1} \mathit {c1}}+\frac {\delta }{\mathit {a1}}} \textit {\_F1} \left (\frac {\left (\mathit {b1} y +\mathit {b0} \right ) \left (\mathit {a1} x +\mathit {a0} \right )^{-\frac {\mathit {b1}}{\mathit {a1}}}}{\mathit {b1}}, \frac {\left (\mathit {c1} z +\mathit {c0} \right ) \left (\mathit {a1} x +\mathit {a0} \right )^{-\frac {\mathit {c1}}{\mathit {a1}}}}{\mathit {c1}}\right ) {\mathrm e}^{\frac {\alpha x}{\mathit {a1}}+\frac {\left (\mathit {b1} y +\mathit {b0} \right ) \beta }{\mathit {b1}^{2}}+\frac {\left (\mathit {c1} z +\mathit {c0} \right ) \gamma }{\mathit {c1}^{2}}}\]
____________________________________________________________________________________