Added April 14, 2019.
Problem Chapter 6.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 x y+b_2 x^2+c_2 x) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*x*y+b2*x^2+c2*x)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-2 \text {a1} \text {b2} x^3-3 \text {a1} \text {c2} x^2+6 \text {a1} z+2 \text {a2} \text {b1} x^3+3 \text {a2} \text {c1} x^2-6 \text {a2} y}{6 \text {a1}},\frac {e^{-\frac {\text {a1} x^2}{2}} (\text {a1} y+\text {b1} x+\text {c1})}{\text {a1}}-\frac {\sqrt {\frac {\pi }{2}} \text {b1} \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\text {a1}^{3/2}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*x*y+b2*x^2+c2*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-\frac {\pi \sqrt {2}\, \mathit {b1} \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )}{2}+\sqrt {\pi }\, \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}}}{\sqrt {\pi }\, \mathit {a1}^{\frac {3}{2}}}, -\frac {-\frac {3 \sqrt {2}\, \left (\frac {\sqrt {\pi }\, \sqrt {\frac {\mathit {a1}}{\pi }}}{\sqrt {\mathit {a1}}}-1\right ) \mathit {a2} \mathit {b1} \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right ) {\mathrm e}^{\frac {\mathit {a1} \,x^{2}}{2}}}{2}+\frac {3 \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) \sqrt {\frac {\mathit {a1}}{\pi }}\, \mathit {a2} \,{\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}} {\mathrm e}^{\frac {\mathit {a1} \,x^{2}}{2}}}{\sqrt {\mathit {a1}}}+\sqrt {\frac {\mathit {a1}}{\pi }}\, \left (-\left (\mathit {b1} x +\frac {3 \mathit {c1}}{2}\right ) \mathit {a1} \mathit {a2} \,x^{2}-3 \mathit {a2} \mathit {b1} x +\left (\mathit {b2} \,x^{3}+\frac {3}{2} \mathit {c2} \,x^{2}-3 z \right ) \mathit {a1}^{2}\right )}{3 \sqrt {\frac {\mathit {a1}}{\pi }}\, \mathit {a1}^{2}}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 x z+b_2 x^2+c_2 x) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*x*z+b2*x^2+c2*x)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\frac {\text {a1} x^2}{2}} (\text {a1} y+\text {b1} x+\text {c1})}{\text {a1}}-\frac {\sqrt {\frac {\pi }{2}} \text {b1} \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\text {a1}^{3/2}},\frac {e^{-\frac {\text {a2} x^2}{2}} (\text {a2} z+\text {b2} x+\text {c2})}{\text {a2}}-\frac {\sqrt {\frac {\pi }{2}} \text {b2} \text {erf}\left (\frac {\sqrt {\text {a2}} x}{\sqrt {2}}\right )}{\text {a2}^{3/2}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*x*z+b2*x^2+c2*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-\frac {\pi \sqrt {2}\, \mathit {b1} \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )}{2}+\sqrt {\pi }\, \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}}}{\sqrt {\pi }\, \mathit {a1}^{\frac {3}{2}}}, \frac {-\frac {\pi \sqrt {2}\, \mathit {b2} \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a2}}\, x}{2}\right )}{2}+\sqrt {\pi }\, \left (\mathit {a2}^{\frac {3}{2}} z +\left (\mathit {b2} x +\mathit {c2} \right ) \sqrt {\mathit {a2}}\right ) {\mathrm e}^{-\frac {\mathit {a2} \,x^{2}}{2}}}{\sqrt {\pi }\, \mathit {a2}^{\frac {3}{2}}}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 y z+b_2 y^2+c_2 y) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*y*z+b2*y^2+c2*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*y*z+b2*y^2+c2*y)*diff(w(x,y,z),z)= 0; 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 {-\frac {\pi \sqrt {2}\, \mathit {b1} \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )}{2}+\sqrt {\pi }\, \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}}}{\sqrt {\pi }\, \mathit {a1}^{\frac {3}{2}}}, z \,{\mathrm e}^{-\frac {\int _{}^{x}\frac {2 \left (-\frac {\sqrt {2}\, \pi \left (-\erf \left (\frac {\sqrt {2}\, \textit {\_b} \sqrt {\mathit {a1}}}{2}\right )+\erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )\right ) \mathit {b1} \,{\mathrm e}^{\frac {\textit {\_b}^{2} \mathit {a1}}{2}}}{2}+\sqrt {\pi }\, \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) {\mathrm e}^{\frac {\textit {\_b}^{2} \mathit {a1}}{2}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}}-\sqrt {\pi }\, \left (\textit {\_b} \mathit {b1} +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) \mathit {a2}}{\mathit {a1}^{\frac {3}{2}}}d \textit {\_b}}{2 \sqrt {\pi }}}-\frac {\int _{}^{x}\frac {2 \left (\frac {\pi ^{\frac {3}{2}} \sqrt {\mathit {a1}}\, \mathit {b1}^{2} \mathit {b2} \erf \left (\frac {\sqrt {2}\, \textit {\_b} \sqrt {\mathit {a1}}}{2}\right )^{2} {\mathrm e}^{\textit {\_b}^{2} \mathit {a1}}}{2}-\pi ^{\frac {3}{2}} \sqrt {\mathit {a1}}\, \mathit {b1}^{2} \mathit {b2} \erf \left (\frac {\sqrt {2}\, \textit {\_b} \sqrt {\mathit {a1}}}{2}\right ) \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right ) {\mathrm e}^{\textit {\_b}^{2} \mathit {a1}}+\frac {\pi ^{\frac {3}{2}} \sqrt {\mathit {a1}}\, \mathit {b1}^{2} \mathit {b2} \erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )^{2} {\mathrm e}^{\textit {\_b}^{2} \mathit {a1}}}{2}-\left (-\erf \left (\frac {\sqrt {2}\, \textit {\_b} \sqrt {\mathit {a1}}}{2}\right )+\erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )\right ) \sqrt {2}\, \pi \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) \sqrt {\mathit {a1}}\, \mathit {b1} \mathit {b2} \,{\mathrm e}^{\textit {\_b}^{2} \mathit {a1}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}}+\sqrt {\pi }\, \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right )^{2} \sqrt {\mathit {a1}}\, \mathit {b2} \,{\mathrm e}^{\textit {\_b}^{2} \mathit {a1}} {\mathrm e}^{-\mathit {a1} \,x^{2}}+\left (-\frac {\pi \sqrt {2}\, \left (-\erf \left (\frac {\sqrt {2}\, \textit {\_b} \sqrt {\mathit {a1}}}{2}\right )+\erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )\right ) \mathit {b1}}{2}+\sqrt {\pi }\, \left (\mathit {a1}^{\frac {3}{2}} y +\left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}}\right ) \left (-2 \textit {\_b} \mathit {b1} \mathit {b2} +\mathit {c2} \mathit {a1} -2 \mathit {b2} \mathit {c1} \right ) \mathit {a1} \,{\mathrm e}^{\frac {\textit {\_b}^{2} \mathit {a1}}{2}}-\left (\mathit {a1}^{\frac {5}{2}} \mathit {c2} -\left (\textit {\_b} \mathit {b1} +\mathit {c1} \right ) \mathit {a1}^{\frac {3}{2}} \mathit {b2} \right ) \sqrt {\pi }\, \left (\textit {\_b} \mathit {b1} +\mathit {c1} \right )\right ) {\mathrm e}^{-\frac {\mathit {a2} \left (\int \left (\left (-\sqrt {\pi }\, \sqrt {2}\, \left (-\erf \left (\frac {\sqrt {2}\, \textit {\_b} \sqrt {\mathit {a1}}}{2}\right )+\erf \left (\frac {\sqrt {2}\, \sqrt {\mathit {a1}}\, x}{2}\right )\right ) \mathit {b1} +\left (2 \mathit {a1}^{\frac {3}{2}} y +2 \left (\mathit {b1} x +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{2}}{2}}\right ) {\mathrm e}^{\frac {\textit {\_b}^{2} \mathit {a1}}{2}}-2 \left (\textit {\_b} \mathit {b1} +\mathit {c1} \right ) \sqrt {\mathit {a1}}\right )d \textit {\_b} \right )}{2 \mathit {a1}^{\frac {3}{2}}}}}{\mathit {a1}^{\frac {7}{2}}}d \textit {\_b}}{2 \sqrt {\pi }}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x +(a_1 x y+b_1 y^2) w_y + (a_2 x z+b_2 z^2) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x+b1*y^2)*D[w[x, y,z], y] +(a2*x*z+b2*z^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {2 \left (\text {b1} x y J_{\frac {1}{3}}\left (\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )+\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} J_{-\frac {2}{3}}\left (\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )\right )}{(2 \text {b1} x y+1) J_{-\frac {1}{3}}\left (\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )+\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} J_{-\frac {4}{3}}\left (\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )-\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} J_{\frac {2}{3}}\left (\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )},\frac {\sqrt {\frac {\pi }{2}} \text {b2} \operatorname {Erfi}\left (\frac {\sqrt {\text {a2}} x}{\sqrt {2}}\right )}{\sqrt {\text {a2}}}+\frac {e^{\frac {\text {a2} x^2}{2}}}{z}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x+b1*y^2)*diff(w(x,y,z),y)+ (a2*x*z+b2*z^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\mathit {b1} y \AiryBi \left (-\left (\mathit {a1} \mathit {b1} \right )^{\frac {1}{3}} x \right )-\left (\mathit {a1} \mathit {b1} \right )^{\frac {1}{3}} \AiryBi \left (1, -\left (\mathit {a1} \mathit {b1} \right )^{\frac {1}{3}} x \right )}{-\mathit {b1} y \AiryAi \left (-\left (\mathit {a1} \mathit {b1} \right )^{\frac {1}{3}} x \right )+\left (\mathit {a1} \mathit {b1} \right )^{\frac {1}{3}} \AiryAi \left (1, -\left (\mathit {a1} \mathit {b1} \right )^{\frac {1}{3}} x \right )}, \frac {\sqrt {\pi }\, \mathit {b2} z \erf \left (\frac {\sqrt {-2 \mathit {a2}}\, x}{2}\right )+\sqrt {-2 \mathit {a2}}\, {\mathrm e}^{\frac {\mathit {a2} \,x^{2}}{2}}}{\sqrt {-2 \mathit {a2}}\, z}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + x z w_y - x y w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] +x*z*D[w[x, y,z], y] -x*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \sin \left (\frac {x^2}{2 a}\right )+z \cos \left (\frac {x^2}{2 a}\right ),y \cos \left (\frac {x^2}{2 a}\right )-z \sin \left (\frac {x^2}{2 a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+x*z*diff(w(x,y,z),y)- x*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (y^{2}+z^{2}, -2 a \arctan \left (\frac {y}{z}\right )+x^{2}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ c x w_x + c y w_y +(a x^2+b y^2) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = c*x*D[w[x, y,z], x] +c*y*D[w[x, y,z], y] +(a*x^2+b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},-\frac {a x^2+b y^2-2 c z}{2 c}\right )\right \}\right \}\]
Maple ✓
restart; pde := c*x*diff(w(x,y,z),x)+c*y*diff(w(x,y,z),y)+(a*x^2+b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {y}{x}, \frac {-a \,x^{2}-b \,y^{2}+2 c z}{2 c}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ c z w_x -a(2 a x-b)y w_y +a (2 a x-b)z w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = c*z*D[w[x, y,z], x] -a*(2*a*x-b)*y*D[w[x, y,z], y] +a*(2*a*x-b)*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (c y z,\frac {-a^2 x^2+a b x+c z}{c}\right )\right \}\right \}\]
Maple ✓
restart; pde := c*z*diff(w(x,y,z),x)-a*(2*a*x-b)*diff(w(x,y,z),y)+a*(2*a*x-b)*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-a^{2} x^{2}+a x b +c z}{c}, y +\ln \left (c z \right )\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a c x^2 w_x -a c x y w_y -b^2 y^2 w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*c*x^2*D[w[x, y,z], x] -a*c*x*y*D[w[x, y,z], y] -b^2*y^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (x y,z-\frac {b^2 y^2}{3 a c x}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*c*x^2*diff(w(x,y,z),x) -a*c*x*y*diff(w(x,y,z),y)-b^2*y^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (x y , \frac {3 z a c \,x^{3}-b^{2} x^{2} y^{2}}{3 a c \,x^{3}}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2 w_x +b y^2 w_y +c z^2 w_z= 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a x -b y}{a x y}, \frac {a x -c z}{a x z}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a b x^2 w_x +c z^2 w_y +2 a b x z w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*b*x^2*D[w[x, y,z], x] +c*z^2*D[w[x, y,z], y] +2*a*b*x*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {z}{x^2},y-\frac {c z^2}{3 a b x}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*b*x^2*diff(w(x,y,z),x) +c*z^2*diff(w(x,y,z),y)+2*a*b*x*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {z}{x^{2}}, \frac {3 a x y b -c \,z^{2}}{3 a b x}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c x y w_x +a^2 c x^2 w_y - b y (2 a x+c z)w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*c*x*y*D[w[x, y,z], x] +a^2*c*x^2*D[w[x, y,z], y] -b*y*(2*a*x+c*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b y^2-a^2 x^2}{2 b},\frac {x (a x+c z)}{c}\right )\right \}\right \}\]
Maple ✓
restart; pde := b*c*x*y*diff(w(x,y,z),x) +a^2*c*x^2*diff(w(x,y,z),y)-b*y*(2*a*x+c*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-a^{2} x^{2}+b \,y^{2}}{b}, \frac {\left (a x +c z \right ) x}{c}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c x y w_x +c^2 y z w_y + b^2 y^2 w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*c*x*y*D[w[x, y,z], x] +c^2*y*z*D[w[x, y,z], y] +b^2*y^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b y+c z}{2 b x},\frac {x (b y-c z)}{2 b}\right )\right \}\right \}\]
Maple ✓
restart; pde := b*c*x*y*diff(w(x,y,z),x) +c^2*y*z*diff(w(x,y,z),y)+b^2*y^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-b^{2} y^{2}+c^{2} z^{2}}{c^{2}}, x \left (b y \,\mathrm {csgn}\left (b \right )+c z \right )^{-\mathrm {csgn}\left (b \right )}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x y w_x +y(y-a)w_y +z(y-a) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*y*D[w[x, y,z], x] +y*(y-a)*D[w[x, y,z], y] +z*(y-a)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y-a}{x},\frac {z}{y}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*y*diff(w(x,y,z),x) +y*(y-a)*diff(w(x,y,z),y)+z*(y-a)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-a +y}{x}, \frac {z}{y}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b y^2 w_x -a x y w_y +c x z w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*y^2*D[w[x, y,z], x] -a*x*y*D[w[x, y,z], y] +c*x*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a x^2+b y^2}{2 b},z \left (-b y^2\right )^{\frac {c}{2 a}}\right )\right \}\right \}\]
Maple ✓
restart; pde := b*y^2*diff(w(x,y,z),x) -a*x*y*diff(w(x,y,z),y)+c*x*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \,x^{2}+b \,y^{2}}{b}, z \left (-b \,y^{2}\right )^{\frac {c}{2 a}}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ c x z w_x + 2 a x y w_y -(2 a x+c z) z w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = c*x*z*D[w[x, y,z], x] +2*a*x*y*D[w[x, y,z], y] -(2*a*x+c*z)*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-c x y z,x \left (\frac {a x}{c}+z\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := c*x*z*diff(w(x,y,z),x) +2*a*x*y*diff(w(x,y,z),y)-(2*a*x+c*z)*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\left (a x +c z \right ) x}{c}, -c x y z \right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.16, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ c x z w_x + c y z w_y +a b x y w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = c*x*z*D[w[x, y,z], x] +c*y*z*D[w[x, y,z], y] +a*b*x*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {c z^2-a b x y}{2 c}\right )\right \}\right \}\]
Maple ✓
restart; pde := c*x*z*diff(w(x,y,z),x) +c*y*z*diff(w(x,y,z),y)+a*b*x*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {y}{x}, \frac {-a x y b +c \,z^{2}}{c}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.17, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ c x z w_x - c y z w_y +(b y^2-a x) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = c*x*z*D[w[x, y,z], x] -c*y*z*D[w[x, y,z], y] +(b*y^2-a*x)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (x y,\frac {2 a x+b y^2+c z^2}{2 c}\right )\right \}\right \}\]
Maple ✓
restart; pde := c*x*z*diff(w(x,y,z),x)-c*y*z*diff(w(x,y,z),y)+(b*y^2-a*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (x y , \frac {b \,x^{2} y^{2}+z^{2} c \,x^{2}+2 a \,x^{3}}{c \,x^{2}}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.18, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ c x z w_x - c y z w_y +(a x^2+b y^2 ) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = c*x*z*D[w[x, y,z], x] -c*y*z*D[w[x, y,z], y] +(a*x^2+b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (x y,\frac {-a x^2+b y^2+c z^2}{2 c}\right )\right \}\right \}\]
Maple ✓
restart; pde := c*x*z*diff(w(x,y,z),x)-c*y*z*diff(w(x,y,z),y)+(a*x^2+b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (x y , \frac {-a \,x^{2}+b \,y^{2}+c \,z^{2}}{c}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.19, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x z w_x + y z w_y +(a x^2+a y^2+ b z^2 ) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*z*D[w[x, y,z], x] +y*z*D[w[x, y,z], y] +(a*x^2+a*y^2+b*z^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {x^{-2 b} \left (a \left (x^2+y^2\right )+(b-1) z^2\right )}{b-1}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*z*diff(w(x,y,z),x)+y*z*diff(w(x,y,z),y)+(a*x^2+a*y^2+b*z^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {y}{x}, \frac {\left (\left (b -1\right ) z^{2}+\left (x^{2}+y^{2}\right ) a \right ) x^{-2 b}}{b -1}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.20, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ 2 c x z w_x + 2 c y z w_y +(c z^2-a x^2- b y^2 ) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = 2*c*x*z*D[w[x, y,z], x] +2*c*y*z*D[w[x, y,z], y] +(c*z^2-a*x^2-b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {a x^2+b y^2+c z^2}{c x}\right )\right \}\right \}\]
Maple ✓
restart; pde := 2*c*x*z*diff(w(x,y,z),x)+2*c*y*z*diff(w(x,y,z),y)+(c*z^2-a*x^2-b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {y}{x}, \frac {a \,x^{2}+b \,y^{2}+c \,z^{2}}{c x}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.21, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c y z w_x + a c x z w_y + a b x y w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*c*y*z*D[w[x, y,z], x] +a*c*x*z*D[w[x, y,z], y] +a*b*x*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b y^2-a x^2}{2 b},\frac {c z^2-a x^2}{2 c}\right )\right \}\right \}\]
Maple ✓
restart; pde := b*c*y*z*diff(w(x,y,z),x)+a*c*x*z*diff(w(x,y,z),y)+a*b*x*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-a \,x^{2}+b \,y^{2}}{b}, \frac {-a \,x^{2}+c \,z^{2}}{c}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.22, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c (x^2-a^2) w_x + c(b x y+a c z ) w_y + b(c x z + a b y) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*c*(x^2-a^2)*D[w[x, y,z], x] +c*(b*x*y+a*c*z)*D[w[x, y,z], y] +b*(c*x*z + a*b*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a c z+b x y}{a^2 b-b x^2},\frac {a (a b y+c x z)}{b \left (a^2-x^2\right )}\right )\right \}\right \}\]
Maple ✗
restart; pde := b*c*(x^2-a^2)*diff(w(x,y,z),x)+c*(b*x*y+a*c*z)*diff(w(x,y,z),y)+b*(c*x*z + a*b*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.23, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b x (b y +c) w_x + (b^2 y^2-a c x ) w_y + b^2 y z w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = b*x*(b*y+c)*D[w[x, y,z], x] + (b^2*y^2-a*c*x )*D[w[x, y,z], y] + b^2*y*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := b*x*(b*y+c)*diff(w(x,y,z),x)+(b^2*y^2-a*c*x )*diff(w(x,y,z),y)+b^2*y*z*diff(w(x,y,z),z)= 0; 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 {b y +c +\left (a x +b y \right ) \ln \left (\frac {-9 a x +9 c}{2 b y +2 c}\right )+\left (-a x -b y \right ) \ln \left (-\frac {9 \left (a x +b y \right ) \left (a x -c \right )}{\left (b y +c \right ) x}\right )}{3 a x +3 b y}, z \,{\mathrm e}^{-\frac {\left (\int _{}^{x}\frac {9 \textit {\_a} a +2 c \,{\mathrm e}^{\RootOf \left (2 \textit {\_Z} a x \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} b y \,{\mathrm e}^{\textit {\_Z}}-2 a x \,{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -c \right )}{\textit {\_a}}\right )-2 a x \,{\mathrm e}^{\textit {\_Z}} \ln \left (-\frac {9 \left (a x -c \right )}{2 \left (b y +c \right )}\right )+2 a x \,{\mathrm e}^{\textit {\_Z}} \ln \left (-\frac {9 \left (a x +b y \right ) \left (a x -c \right )}{\left (b y +c \right ) x}\right )-2 b y \,{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -c \right )}{\textit {\_a}}\right )-2 b y \,{\mathrm e}^{\textit {\_Z}} \ln \left (-\frac {9 \left (a x -c \right )}{2 \left (b y +c \right )}\right )+2 b y \,{\mathrm e}^{\textit {\_Z}} \ln \left (-\frac {9 \left (a x +b y \right ) \left (a x -c \right )}{\left (b y +c \right ) x}\right )-9 \textit {\_Z} a x -9 \textit {\_Z} b y +9 a x \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -c \right )}{\textit {\_a}}\right )+9 a x \ln \left (-\frac {9 \left (a x -c \right )}{2 \left (b y +c \right )}\right )-9 a x \ln \left (-\frac {9 \left (a x +b y \right ) \left (a x -c \right )}{\left (b y +c \right ) x}\right )-2 b y \,{\mathrm e}^{\textit {\_Z}}+9 b y \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (\textit {\_a} a -c \right )}{\textit {\_a}}\right )+9 b y \ln \left (-\frac {9 \left (a x -c \right )}{2 \left (b y +c \right )}\right )-9 b y \ln \left (-\frac {9 \left (a x +b y \right ) \left (a x -c \right )}{\left (b y +c \right ) x}\right )-9 a x -2 c \,{\mathrm e}^{\textit {\_Z}}+9 c \right )}-9 c}{\left (\textit {\_a} a -c \right ) \textit {\_a}}d \textit {\_a} \right )}{9}}\right )\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.24, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x (b y -c z) w_x + y(c z-a x) w_y + z(a x - b y) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*(b*y -c*z)*D[w[x, y,z], x] + y*(c*z-a*x)*D[w[x, y,z], y] + z*(a*x - b*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x y z}{b},\frac {a x+b y+c z}{c}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*(b*y -c*z)*diff(w(x,y,z),x)+ y*(c*z-a*x)*diff(w(x,y,z),y)+z*(a*x - b*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right ) = \frac {c_{4} c_{5} x^{c_{2}} y^{c_{2}} z^{c_{2}} {\mathrm e}^{c_{2}} {\mathrm e}^{-c_{1} x} {\mathrm e}^{-\frac {c_{1} b y}{a}} {\mathrm e}^{-\frac {c_{1} c z}{a}}}{c_{3}}\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.25, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a(y+\beta )(z+\gamma ) w_x -b(x+\alpha )(z+\gamma ) w_y - c(x+\alpha )(y+\beta ) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*(y+beta)*(z+gamma)*D[w[x, y,z], x] -b*(x+alpha)*(z+gamma)*D[w[x, y,z], y] - c*(x+alpha)*(y+beta)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a y (2 \beta +y)+2 \alpha b x+b x^2}{2 a},\frac {a z (2 \gamma +z)+2 \alpha c x+c x^2}{2 a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*(y+beta)*(z+gamma)*diff(w(x,y,z),x)-b*(x+alpha)*(z+gamma)*diff(w(x,y,z),y)- c*(x+alpha)*(y+beta)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right ) = c_{2} c_{4} c_{5} {\mathrm e}^{c_{1} \beta y} {\mathrm e}^{c_{3} \alpha x} {\mathrm e}^{\frac {c_{3} \gamma a z}{c}} {\mathrm e}^{\frac {c_{1} y^{2}}{2}} {\mathrm e}^{\frac {c_{3} x^{2}}{2}} {\mathrm e}^{-\frac {c_{1} b \,z^{2}}{2 c}} {\mathrm e}^{\frac {c_{3} a \,z^{2}}{2 c}} {\mathrm e}^{-\frac {c_{1} \gamma b z}{c}}\]
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.26, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c (a c x z + b^2 y^2) w_x +a c (b c y z-2 a^2 x^2)w_y - a b (2 a b x y+c^2 z^2) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = b*c*(a*c*x*z + b^2*y^2)*D[w[x, y,z], x] +a*c*(b*c*y*z-2*a^2*x^2)*D[w[x, y,z], y] - a*b*(2*a*b*x*y+c^2*z^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := b*c*(a*c*x*z + b^2*y^2)*diff(w(x,y,z),x)+a*c*(b*c*y*z-2*a^2*x^2)*diff(w(x,y,z),y)- a*b*(2*a*b*x*y+c^2*z^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added April 14, 2019.
Problem Chapter 6.2.2.27, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a(y^2+z^2) w_x +x(b z-a y)w_y -x(b y + a z) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*(y^2+z^2)*D[w[x, y,z], x] +x*(b*z-a*y)*D[w[x, y,z], y] -x*(b*y + a*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := a*(y^2+z^2)*diff(w(x,y,z),x)+x*(b*z-a*y)*diff(w(x,y,z),y)-x*(b*y + a*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right ) = c_{1} \textit {\_F5} \left (-\frac {-2 a \arctan \left (\frac {z}{y}\right )+b \ln \left (y^{2}+z^{2}\right )}{2 b}\right ) {\mathrm e}^{\frac {x^{2} \textit {\_c}_{1}}{2}} {\mathrm e}^{-a \textit {\_c}_{1} \left (\int _{}^{y}-\frac {2 \textit {\_a}}{a \cos \left (2 \RootOf \left (2 \textit {\_Z} a -2 a \arctan \left (\frac {z}{y}\right )-b \ln \left (\frac {2 \textit {\_a}^{2}}{\cos \left (2 \textit {\_Z} \right )+1}\right )+b \ln \left (y^{2}+z^{2}\right )\right )\right )-b \sin \left (2 \RootOf \left (2 \textit {\_Z} a -2 a \arctan \left (\frac {z}{y}\right )-b \ln \left (\frac {2 \textit {\_a}^{2}}{\cos \left (2 \textit {\_Z} \right )+1}\right )+b \ln \left (y^{2}+z^{2}\right )\right )\right )+a}d \textit {\_a} \right )}\]
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Added April 14, 2019.
Problem Chapter 6.2.2.28, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b(b y + c z)^2 w_x - a x(b y + 2 c z)w_y +a b x z w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b*(b*y + c*z)^2*D[w[x, y,z], x] - a*x*(b*y + 2*c*z)*D[w[x, y,z], y] +a*b*x*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\begin {align*} & \left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+c^2 z^2\right )}{b},\log (z (b y+c z))\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+c^2 z^2\right )}{b},\log (z (c z-b y))\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+(b y-c z)^2\right )}{b},\log (z (c z-b y))\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {2 \left (a x^2+(b y+c z)^2\right )}{b},\log (z (b y+c z))\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := b*(b*y + c*z)^2*diff(w(x,y,z),x)- a*x*(b*y + 2*c*z)*diff(w(x,y,z),y)+a*b*x*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right ) = c_{1} \textit {\_F5} \left (\frac {\left (b y +c z \right ) z}{b}\right ) {\mathrm e}^{\frac {x^{2} \textit {\_c}_{1}}{2}} {\mathrm e}^{\frac {b^{2} y^{2} \textit {\_c}_{1}}{2 a}} {\mathrm e}^{\frac {b c y z \textit {\_c}_{1}}{2 a}}\]
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Added April 14, 2019.
Problem Chapter 6.2.2.29, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (f_0 x - f_1) w_x + (f_0 y-f_2) w_y + (f_0 z -f_3) w_z= 0 \] Where \[ f_n = a_n + b_n x + c_n y+ d_n z \]
Mathematica ✗
ClearAll["Global`*"]; f[n_]:= a[n] + b[n]*x + c[n]*y+ d[n]*z; pde = (f[0]*x - f[1])*D[w[x, y,z], x] +(f[0]*y-f[2])*D[w[x, y,z], y] +(f[0]*z -f[3])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; f:= n -> a[n] + b[n]*x + c[n]*y+ d[n]*z; pde := (f(0)*x - f(1))*diff(w(x,y,z),x)+(f(0)*y-f(2))*diff(w(x,y,z),y)+(f(0)*z -f(3))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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