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]];
Failed
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( n+1 \right ) {y}^{-m+1}+{x}^{n+1}a \left ( m-1 \right ) }{n+1}},b \left ( \lambda -1 \right ) \int ^{x}\!{{\it \_a}}^{\nu } \left ( \left ( {\frac { \left ( n+1 \right ) {y}^{-m+1}+a \left ( {x}^{n+1}-{{\it \_a}}^{n+1} \right ) \left ( m-1 \right ) }{n+1}} \right ) ^{- \left ( m-1 \right ) ^{-1}} \right ) ^{\mu }{d{\it \_a}}+{z}^{1-\lambda } \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},-\int _1^x\frac {\left ((-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} \text {a2} \text {b1} e^{-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right ) \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}}+\text {b2}+\text {b2} \text {n1}\right ) K[1]^{\text {m1}}}{\text {n1}+1}dK[1]+\text {a2} y (-1)^{-\frac {\text {n2}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n2}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {n2}+1}{\text {n1}+1}} e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {n2}+1}{\text {n1}+1},-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+z\right )\right \}\right \}\]
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 ) ={\it \_F1} \left ( {\frac {1}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) } \left ( -{\it b1}\, \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( {\it n1}+1 \right ) ^{2}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{x}^{{\it m1}+1}{\it a1} \right ) \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) + \left ( -{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\,{x}^{-{\it n1}+{\it m1}} \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}y{\it a1}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \right ) \left ( {\it m1}+{\it n1}+2 \right ) \right ) },-\int ^{x}\!{\frac {1}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) } \left ( - \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{{\it \_a}}^{{\it n2}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{x}^{{\it m1}+1}{\it a1} \right ) {{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \left ( {\it n1}+1 \right ) ^{2}{\it a2}\,{\it b1}\, \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {{\it \_a}}^{{\it n2}-{\it n1}+{\it m1}}+{{\it \_a}}^{{\it n2}+{\it m1}+1}{\it a1} \right ) \left ( {\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( {\it n1}+1 \right ) ^{2}{\it a2}\,{\it b1}\, \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) + \left ( -{{\it \_a}}^{{\it n2}}{x}^{-{\it n1}+{\it m1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it a2}\,{\it b1}\, \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{{\it \_a}}^{{\it n2}-{\it n1}+{\it m1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it a2}\,{\it b1}\, \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{\it a1}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it a2}\,{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}y{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\it \_a}}^{{\it n2}}+{{\it \_a}}^{{\it m1}}{\it b2} \right ) \right ) \left ( {\it m1}+{\it n1}+2 \right ) \right ) }{d{\it \_a}}+z \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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (\text {n2}+1)^{\frac {\text {m1}-\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {\text {m1}+1}{\text {n2}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n2}+1},\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}\right )+z e^{-\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]
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 ) ={\it \_F1} \left ( {\frac {1}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) } \left ( -{\it b1}\, \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( {\it n1}+1 \right ) ^{2}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{x}^{{\it m1}+1}{\it a1} \right ) \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) + \left ( -{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\,{x}^{-{\it n1}+{\it m1}} \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}y{\it a1}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \right ) \left ( {\it m1}+{\it n1}+2 \right ) \right ) },{\frac {1}{{\it a2}\, \left ( {\it m1}+2\,{\it n2}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n2}+2 \right ) } \left ( -{\it b2}\, \left ( {\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m1}-{\it n2}-2}{2\,{\it n2}+2}}} \left ( {\it n2}+1 \right ) ^{2}{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \left ( \left ( {\it m1}+{\it n2}+2 \right ) {x}^{-{\it n2}+{\it m1}}+{\it a2}\,{x}^{{\it m1}+1} \right ) \WhittakerM \left ( {\frac {-{\it n2}+{\it m1}}{2\,{\it n2}+2}},{\frac {{\it m1}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) + \left ( -{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{2\,{\it n2}+2}}}} \left ( {\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) ^{{\frac {-{\it m1}-{\it n2}-2}{2\,{\it n2}+2}}}{\it b2}\,{x}^{-{\it n2}+{\it m1}} \left ( {\it n2}+1 \right ) \left ( {\it m1}+{\it n2}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n2}+2}{2\,{\it n2}+2}},{\frac {{\it m1}+2\,{\it n2}+3}{2\,{\it n2}+2}},{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}} \right ) +{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}z{\it a2}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n2}+3 \right ) \right ) \left ( {\it m1}+{\it n2}+2 \right ) \right ) } \right ) \]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right )+y e^{-\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}},z e^{-\frac {\text {a2} x^{\text {n2}+1}}{\text {n2}+1}}-\int _1^x\text {b2} e^{-\frac {\text {a2} K[1]^{\text {n2}+1}}{\text {n2}+1}} \left (\text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} e^{\frac {\text {a1} \left (K[1]^{\text {n1}+1}-x^{\text {n1}+1}\right )}{\text {n1}+1}} (\text {n1}+1)^{-\frac {\text {n1}}{\text {n1}+1}-1} \left (\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}\right ) (\text {n1}+1)^{\frac {\text {m1}+\text {n1}+1}{\text {n1}+1}}+\left (\text {a1}^{\frac {\text {m1}+1}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {n1}}{\text {n1}+1}} y-\text {b1} e^{\frac {\text {a1} x^{\text {n1}+1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},\frac {\text {a1} K[1]^{\text {n1}+1}}{\text {n1}+1}\right )\right ) (\text {n1}+1)\right )\right )^{\text {m1}}dK[1]\right )\right \}\right \}\]
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 ) ={\it \_F1} \left ( {\frac {1}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) } \left ( -{\it b1}\, \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( {\it n1}+1 \right ) ^{2}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{x}^{{\it m1}+1}{\it a1} \right ) \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) + \left ( -{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\,{x}^{-{\it n1}+{\it m1}} \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}y{\it a1}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \right ) \left ( {\it m1}+{\it n1}+2 \right ) \right ) },-\int ^{x}\!{\it b2}\, \left ( {\frac {1}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) } \left ( - \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}+{x}^{{\it m1}+1}{\it a1} \right ) \left ( {\it n1}+1 \right ) ^{2}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{\it b1}\, \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) +{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\it n1}+1 \right ) ^{2} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {{\it \_a}}^{-{\it n1}+{\it m1}}+{{\it \_a}}^{{\it m1}+1}{\it a1} \right ) \left ( {\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\, \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) - \left ( {x}^{-{\it n1}+{\it m1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\, \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) -{{\it \_a}}^{-{\it n1}+{\it m1}}{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{2\,{\it n1}+2}}}} \left ( {\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\, \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}} \right ) -{{\rm e}^{{\frac {{{\it \_a}}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{-{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}y{\it a1}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \right ) \left ( {\it m1}+{\it n1}+2 \right ) \right ) } \right ) ^{{\it m1}}{{\rm e}^{-{\frac {{{\it \_a}}^{{\it n2}+1}{\it a2}}{{\it n2}+1}}}}{d{\it \_a}}+{{\rm e}^{-{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}z \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\text {b1} (-1)^{\frac {\text {n1}-\text {m1}}{\text {n1}+1}} (\text {n1}+1)^{\frac {\text {m1}-\text {n1}}{\text {n1}+1}} \text {a1}^{-\frac {\text {m1}+1}{\text {n1}+1}} (\text {k1}-1)^{\frac {\text {n1}-\text {m1}}{\text {n1}+1}} \text {Gamma}\left (\frac {\text {m1}+1}{\text {n1}+1},-\frac {\text {a1} (\text {k1}-1) x^{\text {n1}+1}}{\text {n1}+1}\right )+y^{1-\text {k1}} e^{\frac {\text {a1} (\text {k1}-1) x^{\text {n1}+1}}{\text {n1}+1}},\text {b2} (-1)^{\frac {\text {n2}-\text {m2}}{\text {n2}+1}} (\text {n2}+1)^{\frac {\text {m2}-\text {n2}}{\text {n2}+1}} \text {a2}^{-\frac {\text {m2}+1}{\text {n2}+1}} (\text {k1}-1)^{\frac {\text {n2}-\text {m2}}{\text {n2}+1}} \text {Gamma}\left (\frac {\text {m2}+1}{\text {n2}+1},-\frac {\text {a2} (\text {k1}-1) x^{\text {n2}+1}}{\text {n2}+1}\right )+z^{1-\text {k1}} e^{\frac {\text {a2} (\text {k1}-1) x^{\text {n2}+1}}{\text {n2}+1}}\right )\right \}\right \}\]
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 ) ={\it \_F1} \left ( {\frac {1}{{\it a1}\, \left ( {\it m1}+2\,{\it n1}+3 \right ) \left ( {\it m1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) } \left ( -{\it b1}\,{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( {\it k1}-1 \right ) }{2\,{\it n1}+2}}}}{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{y}^{{\frac {{\it k1}}{{\it n1}+1}}} \left ( -{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( {\it k1}-1 \right ) }{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}} \left ( \left ( {\it m1}+{\it n1}+2 \right ) {x}^{-{\it n1}+{\it m1}}-{x}^{{\it m1}+1}{\it a1}\, \left ( {\it k1}-1 \right ) \right ) \left ( {\it n1}+1 \right ) ^{2}{y}^{{\frac {{\it k1}\,{\it n1}}{{\it n1}+1}}} \WhittakerM \left ( {\frac {-{\it n1}+{\it m1}}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},-{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( {\it k1}-1 \right ) }{{\it n1}+1}} \right ) - \left ( {\it m1}+{\it n1}+2 \right ) \left ( {x}^{-{\it n1}+{\it m1}}{y}^{{\frac {{\it k1}}{{\it n1}+1}}}{y}^{{\frac {{\it k1}\,{\it n1}}{{\it n1}+1}}}{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}}{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( {\it k1}-1 \right ) }{2\,{\it n1}+2}}}} \left ( -{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( {\it k1}-1 \right ) }{{\it n1}+1}} \right ) ^{{\frac {-{\it m1}-{\it n1}-2}{2\,{\it n1}+2}}}{\it b1}\, \left ( {\it n1}+1 \right ) \left ( {\it m1}+{\it n1}+2 \right ) \WhittakerM \left ( {\frac {{\it m1}+{\it n1}+2}{2\,{\it n1}+2}},{\frac {{\it m1}+2\,{\it n1}+3}{2\,{\it n1}+2}},-{\frac {{x}^{{\it n1}+1}{\it a1}\, \left ( {\it k1}-1 \right ) }{{\it n1}+1}} \right ) -{{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}\,{\it k1}}{{\it n1}+1}}}}{y}^{{\frac {{\it n1}}{{\it n1}+1}}}{y}^{ \left ( {\it n1}+1 \right ) ^{-1}}{\it a1}\, \left ( {\it m1}+1 \right ) \left ( {\it m1}+2\,{\it n1}+3 \right ) \right ) \right ) \left ( {y}^{{\frac {{\it k1}\,{\it n1}}{{\it n1}+1}}} \right ) ^{-1} \left ( {y}^{{\frac {{\it k1}}{{\it n1}+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{x}^{{\it n1}+1}{\it a1}}{{\it n1}+1}}}} \right ) ^{-1}},{\frac {1}{ \left ( {\it m2}+1 \right ) \left ( {\it m2}+{\it n2}+2 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) {\it a2}} \left ( -{\it b2}\,{{\rm e}^{{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( {\it k1}-1 \right ) }{2\,{\it n2}+2}}}}{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}{z}^{{\frac {{\it k1}}{{\it n2}+1}}} \left ( -{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( {\it k1}-1 \right ) }{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}} \left ( \left ( {\it m2}+{\it n2}+2 \right ) {x}^{-{\it n2}+{\it m2}}-{x}^{{\it m2}+1}{\it a2}\, \left ( {\it k1}-1 \right ) \right ) \left ( {\it n2}+1 \right ) ^{2}{z}^{{\frac {{\it k1}\,{\it n2}}{{\it n2}+1}}} \WhittakerM \left ( {\frac {-{\it n2}+{\it m2}}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},-{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( {\it k1}-1 \right ) }{{\it n2}+1}} \right ) - \left ( {\it m2}+{\it n2}+2 \right ) \left ( {x}^{-{\it n2}+{\it m2}}{z}^{{\frac {{\it k1}}{{\it n2}+1}}}{z}^{{\frac {{\it k1}\,{\it n2}}{{\it n2}+1}}}{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}}{{\rm e}^{{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( {\it k1}-1 \right ) }{2\,{\it n2}+2}}}} \left ( -{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( {\it k1}-1 \right ) }{{\it n2}+1}} \right ) ^{{\frac {-{\it m2}-{\it n2}-2}{2\,{\it n2}+2}}}{\it b2}\, \left ( {\it n2}+1 \right ) \left ( {\it m2}+{\it n2}+2 \right ) \WhittakerM \left ( {\frac {{\it m2}+{\it n2}+2}{2\,{\it n2}+2}},{\frac {{\it m2}+2\,{\it n2}+3}{2\,{\it n2}+2}},-{\frac {{x}^{{\it n2}+1}{\it a2}\, \left ( {\it k1}-1 \right ) }{{\it n2}+1}} \right ) -{{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}{\it k1}}{{\it n2}+1}}}}{z}^{{\frac {{\it n2}}{{\it n2}+1}}}{z}^{ \left ( {\it n2}+1 \right ) ^{-1}}{\it a2}\, \left ( {\it m2}+1 \right ) \left ( {\it m2}+2\,{\it n2}+3 \right ) \right ) \right ) \left ( {z}^{{\frac {{\it k1}\,{\it n2}}{{\it n2}+1}}} \right ) ^{-1} \left ( {z}^{{\frac {{\it k1}}{{\it n2}+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{\it a2}\,{x}^{{\it n2}+1}}{{\it n2}+1}}}} \right ) ^{-1}} \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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{x}^{-n+1}b \left ( m-1 \right ) +a \left ( n-1 \right ) {y}^{-m+1}}{a \left ( n-1 \right ) }},{\frac {-{x}^{-n+1}c \left ( L-1 \right ) +{z}^{1-L}a \left ( n-1 \right ) }{a \left ( n-1 \right ) }} \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{x}^{n+1}b \left ( m+1 \right ) +{y}^{m+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }},{\frac {1}{a} \left ( c \left ( L-1 \right ) \int ^{x}\! \left ( \left ( {\frac {-{x}^{n+1}b \left ( m+1 \right ) +{y}^{m+1}a \left ( n+1 \right ) +{{\it \_a}}^{n+1}b \left ( m+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( m+1 \right ) ^{-1}} \right ) ^{-m}{d{\it \_a}}+{z}^{1-L}a \right ) } \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={{\it \_C5}\,{\it \_C4}{x}^{{\frac {{\it \_C1}}{n}}}{{\rm e}^{{\frac {{\it \_C1}}{{n}^{2}}}}} \left ( {y}^{n} \right ) ^{{\frac {{\it \_C1}}{{n}^{2}}}} \left ( {z}^{n} \right ) ^{{\frac {{\it \_C1}}{{n}^{2}}}} \left ( {{\rm e}^{{\frac {{x}^{n}{\it \_C3}}{{n}^{2}}}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{\it \_C2}}{n}}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{y}^{n}{\it \_C3}}{{n}^{2}}}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {{z}^{n}{\it \_C3}}{{n}^{2}}}}} \right ) ^{-1}}\]
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