Added May 18, 2019.
Problem Chapter 6.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+ a x^n y^m w_y + b x^\nu y^\mu z^\lambda w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +a*x^n*y^m*D[w[x, y,z], y] +b*x^nu *y^mu* z^lambda *D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a x^{n+1}}{n+1}-(m-1)^{\frac {1}{m-1}} y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m,-\frac {b (m-1)^{\frac {\mu }{1-m}} x^{\nu +1} \left ((m-1)^{\frac {1}{m-1}} y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m\right )^{\frac {\mu }{1-m}} \left (\frac {(m-1)^{\frac {1}{m-1}} (n+1) y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m}{a x^{n+1}+(m-1)^{\frac {1}{m-1}} (n+1) y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m}\right )^{\frac {\mu }{m-1}} \, _2F_1\left (\frac {\mu }{m-1},\frac {\nu +1}{n+1};\frac {n+\nu +2}{n+1};\frac {a x^{n+1}}{(m-1)^{\frac {1}{m-1}} (n+1) y \left (\frac {(m-1)^{\frac {1}{1-m}}}{y}\right )^m+a x^{n+1}}\right )}{\nu +1}-(\lambda -1)^{\frac {1}{\lambda -1}} z \left (\frac {(\lambda -1)^{\frac {1}{1-\lambda }}}{z}\right )^{\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+a*x^n*y^m*diff(w(x,y,z),y)+b*x^nu*y^mu*z^lambda*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\left (m -1\right ) a \,x^{n +1}+\left (n +1\right ) y^{-m +1}}{n +1}, \left (\lambda -1\right ) b \left (\int _{}^{x}\textit {\_a}^{\nu } \left (\left (\frac {\left (-\textit {\_a}^{n +1}+x^{n +1}\right ) \left (m -1\right ) a +\left (n +1\right ) y^{-m +1}}{n +1}\right )^{-\frac {1}{m -1}}\right )^{\mu }d \textit {\_a} \right )+z^{-\lambda +1}\right )\]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1}) w_y +(a_2 x^{n_2} y + b_2 x^{m_1}) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x^n1*y+b1*x^m1)*D[w[x, y,z], y] +(a2*x^n2*y+b2*x^m1)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},-\int _1^x\frac {\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right ) \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}}+\text {b2}+\text {b2} \text {n1}\right ) K[1]^{\text {m1}}}{\text {n1}+1}dK[1]+\text {a2} y (-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+z\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1)*diff(w(x,y,z),y)+(a2*x^n2*y+b2*x^m1)*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 (\mathit {a1} \,x^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) x^{\mathit {m1} -\mathit {n1}}\right ) \left (\mathit {n1} +1\right )^{2} \mathit {b1} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (-\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {b1} \,x^{\mathit {m1} -\mathit {n1}} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \mathit {a1} y \,{\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}\right ) \left (\mathit {m1} +\mathit {n1} +2\right )}{\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {a1}}, z -\left (\int _{}^{x}\frac {-\left (\mathit {n1} +1\right )^{2} \left (\mathit {a1} \,x^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) x^{\mathit {m1} -\mathit {n1}}\right ) \mathit {a2} \mathit {b1} \,\textit {\_a}^{\mathit {n2}} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {n1} +1\right )^{2} \left (\mathit {a1} \,\textit {\_a}^{\mathit {m1} +\mathit {n2} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) \textit {\_a}^{\mathit {m1} -\mathit {n1} +\mathit {n2}}\right ) \mathit {a2} \mathit {b1} \left (\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (-\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {a2} \mathit {b1} \,\textit {\_a}^{\mathit {n2}} x^{\mathit {m1} -\mathit {n1}} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {a2} \mathit {b1} \,\textit {\_a}^{\mathit {m1} -\mathit {n1} +\mathit {n2}} \left (\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \left (\mathit {a2} y \,\textit {\_a}^{\mathit {n2}} {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}+\mathit {b2} \,\textit {\_a}^{\mathit {m1}}\right ) \mathit {a1} \right ) \left (\mathit {m1} +\mathit {n1} +2\right )}{\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {a1}}d \textit {\_a} \right )\right )\]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1}) w_y +(a_2 x^{n_2} z + b_2 x^{m_1}) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x^n1*y+b1*x^m1)*D[w[x, y,z], y] +(a2*x^n2*z+b2*x^m1)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (\text {n2}+1)^{\frac {\text {m1}-\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {\text {m1}+1}{\text {n2}+1}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n2}+1},\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}\right )+z e^{-\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*x^m1)*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 (\mathit {n1} +1\right )^{2} \left (\mathit {a1} \,x^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) x^{\mathit {m1} -\mathit {n1}}\right ) \mathit {b1} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (-\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {b1} \,x^{\mathit {m1} -\mathit {n1}} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \mathit {a1} y \,{\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}\right ) \left (\mathit {m1} +\mathit {n1} +2\right )}{\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {a1}}, \frac {-\left (\mathit {n2} +1\right )^{2} \left (\mathit {a2} \,x^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n2} +2\right ) x^{\mathit {m1} -\mathit {n2}}\right ) \mathit {b2} \left (\frac {\mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {m1} -\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n2}}{2 \mathit {n2} +2}, \frac {\mathit {m1} +2 \mathit {n2} +3}{2 \mathit {n2} +2}, \frac {\mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{-\frac {\mathit {a2} \,x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+\left (-\left (\mathit {n2} +1\right ) \left (\mathit {m1} +\mathit {n2} +2\right ) \mathit {b2} \,x^{\mathit {m1} -\mathit {n2}} \left (\frac {\mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {m1} -\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n2} +2}{2 \mathit {n2} +2}, \frac {\mathit {m1} +2 \mathit {n2} +3}{2 \mathit {n2} +2}, \frac {\mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{-\frac {\mathit {a2} \,x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n2} +3\right ) \mathit {a2} z \,{\mathrm e}^{-\frac {\mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}}\right ) \left (\mathit {m1} +\mathit {n2} +2\right )}{\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n2} +3\right ) \left (\mathit {m1} +\mathit {n2} +2\right ) \mathit {a2}}\right )\]
____________________________________________________________________________________
Added May 18, 2019.
Problem Chapter 6.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1}) w_y +(a_2 x^{n_2} z + b_2 y^{m_1}) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x^n1*y+b1*x^m1)*D[w[x, y,z], y] +(a2*x^n2*z+b2*y^m1)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},z e^{-\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}}-\int _1^x\text {b2} e^{-\frac {\text {a2} K[1]^{\text {n2}+1}}{\text {n2}+1}} \left (\text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {m1}+\text {n1}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*y^m1)*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 (\mathit {n1} +1\right )^{2} \left (\mathit {a1} \,x^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) x^{\mathit {m1} -\mathit {n1}}\right ) \mathit {b1} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (-\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {b1} \,x^{\mathit {m1} -\mathit {n1}} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \mathit {a1} y \,{\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}\right ) \left (\mathit {m1} +\mathit {n1} +2\right )}{\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {a1}}, z \,{\mathrm e}^{-\frac {\mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}}-\left (\int _{}^{x}\mathit {b2} \left (\frac {-\left (\mathit {n1} +1\right )^{2} \left (\mathit {a1} \,x^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) x^{\mathit {m1} -\mathit {n1}}\right ) \mathit {b1} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {n1} +1\right )^{2} \left (\mathit {a1} \,\textit {\_a}^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) \textit {\_a}^{\mathit {m1} -\mathit {n1}}\right ) \mathit {b1} \left (\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{2 \mathit {n1} +2}}-\left (\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {b1} \,x^{\mathit {m1} -\mathit {n1}} \left (\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}-\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {b1} \,\textit {\_a}^{\mathit {m1} -\mathit {n1}} \left (\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, \frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{2 \mathit {n1} +2}}-\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \mathit {a1} y \,{\mathrm e}^{\frac {\mathit {a1} \,\textit {\_a}^{\mathit {n1} +1}}{\mathit {n1} +1}} {\mathrm e}^{-\frac {\mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}\right ) \left (\mathit {m1} +\mathit {n1} +2\right )}{\left (\mathit {m1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \mathit {a1}}\right )^{\mathit {m1}} {\mathrm e}^{-\frac {\mathit {a2} \,\textit {\_a}^{\mathit {n2} +1}}{\mathit {n2} +1}}d \textit {\_a} \right )\right )\]
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Added May 18, 2019.
Problem Chapter 6.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x+ (a_1 x^{n_1} y + b_1 x^{m_1} y^k1) w_y +(a_2 x^{n_2} z + b_2 x^{m_2} z^{k_2}) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] +(a1*x^n1*y+b1*x^m1*y^k1)*D[w[x, y,z], y] +(a2*x^n2*z+b2*x^m2*z^k1)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (-1)^{\frac {\text {n1}-\text {m1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} (\text {k1}-1)^{\frac {\text {n1}-\text {m1}}{\text {n1}+1}} \operatorname {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},-\frac {\text {a1} (\text {k1}-1) x^{\text {n1}+1}}{\text {n1}+1}\right )+y^{1-\text {k1}} e^{\frac {\text {a1} (\text {k1}-1) x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (-1)^{\frac {\text {n2}-\text {m2}}{\text {n2}+1}} (\text {n2}+1)^{\frac {\text {m2}-\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {\text {m2}+1}{\text {n2}+1}} (\text {k1}-1)^{\frac {\text {n2}-\text {m2}}{\text {n2}+1}} \operatorname {Gamma}\left (\frac {\text {m2}+1}{\text {n2}+1},-\frac {\text {a2} (\text {k1}-1) x^{\text {n2}+1}}{\text {n2}+1}\right )+z^{1-\text {k1}} e^{\frac {\text {a2} (\text {k1}-1) x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+(a1*x^n1*y+b1*x^m1*y^k1)*diff(w(x,y,z),y)+(a2*x^n2*z+b2*x^m2*z^k1)*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 (-\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {m1} +1}+\left (\mathit {m1} +\mathit {n1} +2\right ) x^{\mathit {m1} -\mathit {n1}}\right ) \left (\mathit {n1} +1\right )^{2} \mathit {b1} \left (-\frac {\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} -\mathit {n1}}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {m1} +\mathit {n1} +2\right ) \left (-\left (\mathit {n1} +1\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {b1} \,x^{\mathit {m1} -\mathit {n1}} \left (-\frac {\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right )^{\frac {-\mathit {m1} -\mathit {n1} -2}{2 \mathit {n1} +2}} \WhittakerM \left (\frac {\mathit {m1} +\mathit {n1} +2}{2 \mathit {n1} +2}, \frac {\mathit {m1} +2 \mathit {n1} +3}{2 \mathit {n1} +2}, -\frac {\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}\right ) {\mathrm e}^{\frac {\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{2 \mathit {n1} +2}}+\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \mathit {a1} \,y^{-\mathit {k1} +1} {\mathrm e}^{\frac {\left (\mathit {k1} -1\right ) \mathit {a1} \,x^{\mathit {n1} +1}}{\mathit {n1} +1}}\right )}{\left (\mathit {m1} +1\right ) \left (\mathit {m1} +2 \mathit {n1} +3\right ) \left (\mathit {m1} +\mathit {n1} +2\right ) \mathit {a1}}, \frac {-\left (-\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {m2} +1}+\left (\mathit {m2} +\mathit {n2} +2\right ) x^{\mathit {m2} -\mathit {n2}}\right ) \left (\mathit {n2} +1\right )^{2} \mathit {b2} \left (-\frac {\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {m2} -\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (\frac {\mathit {m2} -\mathit {n2}}{2 \mathit {n2} +2}, \frac {\mathit {m2} +2 \mathit {n2} +3}{2 \mathit {n2} +2}, -\frac {\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{\frac {\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+\left (\mathit {m2} +\mathit {n2} +2\right ) \left (-\left (\mathit {n2} +1\right ) \left (\mathit {m2} +\mathit {n2} +2\right ) \mathit {b2} \,x^{\mathit {m2} -\mathit {n2}} \left (-\frac {\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right )^{\frac {-\mathit {m2} -\mathit {n2} -2}{2 \mathit {n2} +2}} \WhittakerM \left (\frac {\mathit {m2} +\mathit {n2} +2}{2 \mathit {n2} +2}, \frac {\mathit {m2} +2 \mathit {n2} +3}{2 \mathit {n2} +2}, -\frac {\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}\right ) {\mathrm e}^{\frac {\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{2 \mathit {n2} +2}}+\left (\mathit {m2} +1\right ) \left (\mathit {m2} +2 \mathit {n2} +3\right ) \mathit {a2} \,z^{-\mathit {k1} +1} {\mathrm e}^{\frac {\left (\mathit {k1} -1\right ) \mathit {a2} \,x^{\mathit {n2} +1}}{\mathit {n2} +1}}\right )}{\left (\mathit {m2} +1\right ) \left (\mathit {m2} +2 \mathit {n2} +3\right ) \left (\mathit {m2} +\mathit {n2} +2\right ) \mathit {a2}}\right )\]
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Added May 18, 2019.
Problem Chapter 6.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^n w_x+ b y^m w_y +c z^l w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y,z], x] +b*y^m*D[w[x, y,z], y] +c*z^L*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b x^{1-n}}{a (n-1)}-\frac {\left (\frac {1}{y}\right )^{m-1}}{m-1},\frac {c x^{1-n}}{a (n-1)}-\frac {\left (\frac {1}{z}\right )^{L-1}}{L-1}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^n*diff(w(x,y,z),x)+b*y^m*diff(w(x,y,z),y)+c*z^L*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\left (n -1\right ) a \,y^{-m +1}-\left (m -1\right ) b \,x^{-n +1}}{\left (n -1\right ) a}, \frac {\left (n -1\right ) a \,z^{-L +1}-\left (L -1\right ) c \,x^{-n +1}}{\left (n -1\right ) a}\right )\]
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Added May 18, 2019.
Problem Chapter 6.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a y^m w_x+ b x^n w_y +c z^l w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*y^m*D[w[x, y,z], x] +b*x^n*D[w[x, y,z], y] +c*z^L*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^m*diff(w(x,y,z),x)+b*x^n*diff(w(x,y,z),y)+c*z^L*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\left (n +1\right ) a \,y^{m +1}-\left (m +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}, \frac {a \,z^{-L +1}+\left (L -1\right ) c \left (\int _{}^{x}\left (\left (\frac {\left (n +1\right ) a \,y^{m +1}+\left (m +1\right ) b \,\textit {\_a}^{n +1}-\left (m +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {1}{m +1}}\right )^{-m}d \textit {\_a} \right )}{a}\right )\]
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Added May 18, 2019.
Problem Chapter 6.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x(y^n - z^n) w_x+ y(z^n-x^n) w_y +z(x^n-y^n) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*(y^n-z^n)*D[w[x, y,z], x] +y*(z^n-x^n)*D[w[x, y,z], y] +z*(x^n-y^n)*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 := x*(y^n-z^n)*diff(w(x,y,z),x)+y*(z^n-x^n)*diff(w(x,y,z),y)+z*(x^n-y^n)*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'));
\[w \left (x , y , z\right ) = c_{4} c_{5} x^{\frac {c_{1}}{n}} \left (y^{n}\right )^{\frac {c_{1}}{n^{2}}} \left (z^{n}\right )^{\frac {c_{1}}{n^{2}}} {\mathrm e}^{\frac {c_{1}}{n^{2}}} {\mathrm e}^{-\frac {c_{2}}{n}} {\mathrm e}^{-\frac {c_{3} x^{n}}{n^{2}}} {\mathrm e}^{-\frac {c_{3} y^{n}}{n^{2}}} {\mathrm e}^{-\frac {c_{3} z^{n}}{n^{2}}}\]
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