Added July 2, 2019.
Problem Chapter 8.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a x^n w_y + b x^m w_z = (c e^{\lambda x} y + k e^{\beta x} z + s e^{\gamma x}) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*x^n*D[w[x, y,z], y] + b*x^m*D[w[x,y,z],z]== (c*Exp[lambda*x]*y+k*Exp[beta*x]*z+s*Exp[gamma*x])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (-\frac {a c x^n (-\lambda x)^{-n} \operatorname {Gamma}(n+2,-\lambda x)}{\lambda ^2 (n+1)}-\frac {b k x^m (-\beta x)^{-m} \operatorname {Gamma}(m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {c e^{\lambda x} \left (-a x^{n+1}+n y+y\right )}{\lambda (n+1)}+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*x^n*diff(w(x,y,z),y)+c*x^m*diff(w(x,y,z),z)= (c*exp(lambda*x)*y+k*exp(beta*x)*z+s*exp(gamma*x))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a x^{n +1}+\left (n +1\right ) y}{n +1}, \frac {-c x^{m +1}+\left (m +1\right ) z}{m +1}\right ) {\mathrm e}^{\frac {\left (n +1\right ) \left (m +1\right ) \left (\Gamma \left (n \right )-\Gamma \left (n , -\lambda x \right )\right ) a \beta ^{2} c \gamma n x^{n} \left (-\lambda x \right )^{-n}+\left (n +1\right ) \left (m +1\right ) \left (\Gamma \left (m \right )-\Gamma \left (m , -\beta x \right )\right ) c \gamma k \lambda ^{2} m x^{m} \left (-\beta x \right )^{-m}+\left (n +1\right ) \beta c \gamma k \lambda ^{2} x^{m +1}-\left (m +1\right ) \left (-a \beta ^{2} c \gamma \lambda x^{n +1}+\left (n +1\right ) \left (\left (a x^{n}-\lambda y \right ) \beta ^{2} c \gamma \,{\mathrm e}^{\lambda x}+\left (-\left (\beta z -c x^{m}\right ) \gamma k \lambda \,{\mathrm e}^{\beta x}+\left (\beta c \gamma y -\beta \lambda s \,{\mathrm e}^{\gamma x}+\left (\gamma k z +\beta s \right ) \lambda \right ) \beta \right ) \lambda \right )\right )}{\left (m +1\right ) \left (n +1\right ) \beta ^{2} \gamma \lambda ^{2}}}\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\lambda x} w_y + b x^m w_z = (c x^n y + k e^{\beta x} z + s e^{\gamma x}) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] + b*x^m*D[w[x,y,z],z]== (c*x^n*y+k*Exp[beta*x]*z+s*Exp[gamma*x])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (\frac {a c x^n (-\lambda x)^{-n} \operatorname {Gamma}(n+1,-\lambda x)}{\lambda ^2}-\frac {b k x^m (-\beta x)^{-m} \operatorname {Gamma}(m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {c x^{n+1} \left (\lambda y-a e^{\lambda x}\right )}{\lambda (n+1)}+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*x^m*diff(w(x,y,z),z)= (c*x^n*y+k*exp(beta*x)*z+s*exp(gamma*x))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+\lambda y}{\lambda }, \frac {-b x^{m +1}+\left (m +1\right ) z}{m +1}\right ) {\mathrm e}^{\int _{}^{x}\frac {b k \lambda \mathit {\_a}^{m +1} {\mathrm e}^{\mathit {\_a} \beta }-b k \lambda x^{m +1} {\mathrm e}^{\mathit {\_a} \beta }-\left (-a c \mathit {\_a}^{n} {\mathrm e}^{\mathit {\_a} \lambda }-k \lambda z \,{\mathrm e}^{\mathit {\_a} \beta }-\lambda s \,{\mathrm e}^{\mathit {\_a} \gamma }+\left (a \,{\mathrm e}^{\lambda x}-\lambda y \right ) c \mathit {\_a}^{n}\right ) \left (m +1\right )}{\left (m +1\right ) \lambda }d\mathit {\_a}}\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\lambda x} w_y + b y w_z = (k e^{\beta x} z + s e^{\gamma x} ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] + b*y*D[w[x,y,z],z]== (k*Exp[beta*x]*z+s*Exp[gamma*x])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {a b e^{\lambda x} (\lambda x-1)}{\lambda ^2}-b x y+z\right ) \exp \left (\frac {a b k e^{x (\beta +\lambda )}}{\beta ^2 (\beta +\lambda )}+\frac {k e^{\beta x} (\beta z-b y)}{\beta ^2}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= (k*exp(beta*x)*z+s*exp(gamma*x))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+\lambda y}{\lambda }, \frac {\left (\lambda x -1\right ) a b \,{\mathrm e}^{\lambda x}-\left (b x y -z \right ) \lambda ^{2}}{\lambda ^{2}}\right ) {\mathrm e}^{\frac {a b \beta ^{2} \gamma k \,{\mathrm e}^{\left (\beta +\lambda \right ) x}+\left (\beta +\lambda \right ) \left (\beta ^{2} \lambda ^{2} s \,{\mathrm e}^{\gamma x}+\left (\left (-\beta +\lambda \right ) a b \,{\mathrm e}^{\lambda x}-\left (b y -\beta z \right ) \lambda ^{2}\right ) \gamma k \,{\mathrm e}^{\beta x}\right )}{\left (\beta +\lambda \right ) \beta ^{2} \gamma \lambda ^{2}}}\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a y^n w_y + b z^m w_z = (c e^{\lambda x} + k e^{\beta y}+ s e^{\gamma z} ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*y^n*D[w[x, y,z], y] + b*z^m*D[w[x,y,z],z]== (c*Exp[lambda*x]+k*Exp[beta*y]+s*Exp[gamma*z])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-a x-\frac {\left (\frac {1}{y}\right )^{n-1}}{n-1},-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right ) \exp \left (\frac {k \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {n}{n-1}} \left (-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )^n \operatorname {Gamma}\left (1-n,-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \operatorname {Gamma}\left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*y^n*diff(w(x,y,z),y)+b*z^m*diff(w(x,y,z),z)= (c*exp(lambda*x)+k*exp(beta*y)+s*exp(gamma*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\left (\left (n -1\right ) a x y^{n}+y \right ) y^{-n}, \left (\left (m -1\right ) b x z^{m}+z \right ) z^{-m}\right ) {\mathrm e}^{\int _{}^{x}\left (c \,{\mathrm e}^{\mathit {\_a} \lambda }+k \,{\mathrm e}^{\beta \left (\left (\left (-\mathit {\_a} +x \right ) \left (n -1\right ) a y^{n}+y \right ) y^{-n}\right )^{-\frac {1}{n -1}}}+s \,{\mathrm e}^{\gamma \left (\left (\left (-\mathit {\_a} +x \right ) \left (m -1\right ) b z^{m}+z \right ) z^{-m}\right )^{-\frac {1}{m -1}}}\right )d\mathit {\_a}}\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a e^{\beta y} w_y + b z^m w_z = (c e^{\lambda x} + k y^n+ s e^{\gamma z} ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Exp[beta*y]*D[w[x, y,z], y] + b*z^m*D[w[x,y,z],z]== (c*Exp[lambda*x]+k*y^n+s*Exp[gamma*z])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a \beta x+e^{-\beta y}}{\beta },-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right ) \exp \left (-\frac {k \left (-\log \left (e^{-\beta y}\right )\right )^{-n} \left (-\frac {\log \left (e^{-\beta y}\right )}{\beta }\right )^n \operatorname {Gamma}\left (n+1,-\log \left (e^{-\beta y}\right )\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \operatorname {Gamma}\left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*exp(beta*y)*diff(w(x,y,z),y)+b*z^m*diff(w(x,y,z),z)= (c*exp(lambda*x)+k*y^n+s*exp(gamma*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a \beta x -{\mathrm e}^{-\beta y}}{a \beta }, \left (\left (m -1\right ) b x z^{m}+z \right ) z^{-m}\right ) {\mathrm e}^{\int _{}^{x}\left (c \,{\mathrm e}^{\mathit {\_a} \lambda }+k \left (\frac {\ln \left (\frac {1}{\left (-\mathit {\_a} +x \right ) a \beta +{\mathrm e}^{-\beta y}}\right )}{\beta }\right )^{n}+s \,{\mathrm e}^{\gamma \left (\left (\left (-\mathit {\_a} +x \right ) \left (m -1\right ) b z^{m}+z \right ) z^{-m}\right )^{-\frac {1}{m -1}}}\right )d\mathit {\_a}}\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (y^2+ b y+ a e^{\alpha y}(y-b)-b^2) w_y + (z^2+c(x z-1)e^{\beta x}) w_z = k e^{\lambda x} w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (y^2+ b*y+ a*Exp[alpha*y]*(y-b)-b^2)*D[w[x, y,z], y] + (z^2+c*(x*z-1)*Exp[beta*x])*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(y^2+ b*y+ a*exp(alpha*y)*(y-b)-b^2)*diff(w(x,y,z),y)+(z^2+c*(x*z-1)*exp(beta*x))*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right )=c_{1} \mathit {\_F1} \left (x \right ) \mathit {\_F3} \left (z \right ) {\mathrm e}^{\mathit {\_c}_{2} \left (\int \frac {1}{\left (b -y \right ) a \,{\mathrm e}^{\alpha y}+b^{2}-b y -y^{2}}d y \right )}\boldsymbol {\mathrm {where}}\left [\left \{\left (z x -1\right ) c \mathit {\_F1} \left (x \right ) \left (\frac {d}{d z}\mathit {\_F3} \left (z \right )\right ) {\mathrm e}^{\beta x}-k \mathit {\_F1} \left (x \right ) \mathit {\_F3} \left (z \right ) {\mathrm e}^{\lambda x}+z^{2} \mathit {\_F1} \left (x \right ) \left (\frac {d}{d z}\mathit {\_F3} \left (z \right )\right )+\left (-\mathit {\_c}_{2} \mathit {\_F1} \left (x \right )+\frac {d}{d x}\mathit {\_F1} \left (x \right )\right ) \mathit {\_F3} \left (z \right )=0\right \}\right ]\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (y^2+ a e^{\alpha x}(x+1)) w_y + (c e^{\beta x} z^2 + b e^{-\beta x}) w_z = k e^{\lambda x} w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (y^2+ a*Exp[alpha*x]*(x+1))*D[w[x, y,z], y] + (c*Exp[beta*x]*z^2+b*Exp[-beta*x])*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+(y^2+ a*exp(alpha*x)*(x+1))*diff(w(x,y,z),y)+(c*exp(beta*x)*z^2+b*exp(-beta*x))*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a e^{\alpha x}y^2+b e^{-\alpha x}) w_y + (d e^{\beta x} z^2 + c e^{\gamma x}(\gamma -c d e^{(\beta +\gamma )x})) w_z = k e^{\lambda x} w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*Exp[alpha*x]*y^2+b*Exp[-alpha*x])*D[w[x, y,z], y] + (d*Exp[beta*x]*z^2 + c*Exp[gamma*x]*(gamma-c*d*Exp[(beta+gamma)*x]))*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+(a*exp(alpha*x)*y^2+b*exp(-alpha*x))*diff(w(x,y,z),y)+(d*exp(beta*x)*z^2 + c*exp(gamma*x)*(gamma-c*d*exp((beta+gamma)*x)))*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\lambda _1 x} y+b_1 e^{\beta _1 x} y^k) w_y + (a_2 e^{\lambda _2 x} z+b_2 e^{\beta _2 x} z^m) w_z = c x^s w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*Exp[lambda1*x]*y+b1*Exp[beta1*x]*y^k)*D[w[x, y,z], y] + (a2*Exp[lambda2*x]*z+b2*Exp[beta2*x]*z^m)*D[w[x,y,z],z]== c*x^s*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left ((k-1) \int _1^x\text {b1} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {beta1} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {a2} e^{\text {lambda2} K[2]} (m-1)}{\text {lambda2}}+\text {beta2} K[2]}dK[2]+z^{1-m} e^{\frac {\text {a2} (m-1) e^{\text {lambda2} x}}{\text {lambda2}}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a1*exp(lambda1*x)*y+b1*exp(beta1*x)*y^k)*diff(w(x,y,z),y)+(a2*exp(lambda2*x)*z+b2*exp(beta2*x)*z^m)*diff(w(x,y,z),z)= c*x^s*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\left (\left (k -1\right ) \mathit {b1} y^{k} \left (\int {\mathrm e}^{\frac {\beta 1 \lambda 1 x +\left (k -1\right ) \mathit {a1} \,{\mathrm e}^{\lambda 1 x}}{\lambda 1}}d x \right ) {\mathrm e}^{\frac {\mathit {a1} \,{\mathrm e}^{\lambda 1 x}}{\lambda 1}}+y \,{\mathrm e}^{\frac {\mathit {a1} k \,{\mathrm e}^{\lambda 1 x}}{\lambda 1}}\right ) y^{-k} {\mathrm e}^{-\frac {\mathit {a1} \,{\mathrm e}^{\lambda 1 x}}{\lambda 1}}, \left (\left (m -1\right ) \mathit {b2} z^{m} \left (\int {\mathrm e}^{\frac {\beta 2 \lambda 2 x +\left (m -1\right ) \mathit {a2} \,{\mathrm e}^{\lambda 2 x}}{\lambda 2}}d x \right ) {\mathrm e}^{\frac {\mathit {a2} \,{\mathrm e}^{\lambda 2 x}}{\lambda 2}}+z \,{\mathrm e}^{\frac {\mathit {a2} m \,{\mathrm e}^{\lambda 2 x}}{\lambda 2}}\right ) z^{-m} {\mathrm e}^{-\frac {\mathit {a2} \,{\mathrm e}^{\lambda 2 x}}{\lambda 2}}\right ) {\mathrm e}^{\frac {c x^{s +1}}{s +1}}\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\beta _1 x} y+b_1 e^{\gamma _1 x} y^k) w_y + (a_2 e^{\beta _2 x}+b_2 e^{\gamma _2 x+\lambda _2 z} ) w_z = c x^s w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*Exp[beta1*x]*y+b1*Exp[gamma1*x]*y^k)*D[w[x, y,z], y] + (a2*Exp[beta2*x]+b2*Exp[gamma2*x+lambda2*z])*D[w[x,y,z],z]== c*x^s*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a1*exp(beta1*x)*y+b1*exp(gamma1*x)*y^k)*diff(w(x,y,z),y)+ (a2*exp(beta2*x)+b2*exp(gamma2*x+lambda2*z))*diff(w(x,y,z),z)= c*x^s*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\left (\left (k -1\right ) \mathit {b1} y^{k} \left (\int {\mathrm e}^{\frac {\beta 1 \gamma 1 x +\left (k -1\right ) \mathit {a1} \,{\mathrm e}^{\beta 1 x}}{\beta 1}}d x \right ) {\mathrm e}^{\frac {\mathit {a1} \,{\mathrm e}^{\beta 1 x}}{\beta 1}}+y \,{\mathrm e}^{\frac {\mathit {a1} k \,{\mathrm e}^{\beta 1 x}}{\beta 1}}\right ) y^{-k} {\mathrm e}^{-\frac {\mathit {a1} \,{\mathrm e}^{\beta 1 x}}{\beta 1}}, \frac {-\mathit {b2} \lambda 2 \left (\int {\mathrm e}^{\frac {\mathit {a2} \lambda 2 \,{\mathrm e}^{\beta 2 x}}{\beta 2}+\gamma 2 x}d x \right )-{\mathrm e}^{-\frac {\left (-\mathit {a2} \,{\mathrm e}^{\beta 2 x}+\beta 2 z \right ) \lambda 2}{\beta 2}}}{\lambda 2}\right ) {\mathrm e}^{\frac {c x^{s +1}}{s +1}}\]
____________________________________________________________________________________
Added July 2, 2019.
Problem Chapter 8.3.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 x^n+b_1 x^m e^{\lambda y} ) w_y + (a_2 x^k+ b_2 x^r e^{\beta z}) w_z = c x^s w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*x^n+b1*x^m*Exp[lambda*y] )*D[w[x, y,z], y] + (a2*x^k+b2*x^r*Exp[beta*z])*D[w[x,y,z],z]== c*x^s*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left (\frac {(n+1) e^{-\frac {\lambda \left (-\text {a1} x^{n+1}+n y+y\right )}{n+1}}-\text {b1} \lambda x^{m+1} \left (-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \operatorname {Gamma}\left (\frac {m+1}{n+1},-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )}{\text {a1} \text {b1} \lambda ^2 (n+1) (m-n)},\frac {\text {b2} \beta x^{r+1} \left (-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )^{-\frac {r+1}{k+1}} \operatorname {Gamma}\left (\frac {r+1}{k+1},-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\beta \left (-\text {a2} x^{k+1}+k z+z\right )}{k+1}}}{\text {a2} \text {b2} \beta ^2 (k+1) (k-r)}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a1*x^n+b1*x^m*exp(lambda*y) )*diff(w(x,y,z),y)+ (a2*x^k+b2*x^r*exp(beta*z))*diff(w(x,y,z),z)= c*x^s*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {\left (n +1\right )^{2} \left (-\mathit {a1} \lambda x^{m +1}+\left (m +n +2\right ) x^{m -n}\right ) \mathit {b1} \left (-\frac {\mathit {a1} \lambda x^{n +1}}{n +1}\right )^{\frac {-m -n -2}{2 n +2}} \WhittakerM \left (\frac {m -n}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {\mathit {a1} \lambda x^{n +1}}{n +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \lambda x^{n +1}}{2 n +2}}-\left (m +n +2\right ) \left (-\left (n +1\right ) \left (m +n +2\right ) \mathit {b1} x^{m -n} \left (-\frac {\mathit {a1} \lambda x^{n +1}}{n +1}\right )^{\frac {-m -n -2}{2 n +2}} \WhittakerM \left (\frac {m +n +2}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {\mathit {a1} \lambda x^{n +1}}{n +1}\right ) {\mathrm e}^{\frac {\mathit {a1} \lambda x^{n +1}}{2 n +2}}+\left (m +1\right ) \left (m +2 n +3\right ) \mathit {a1} \,{\mathrm e}^{-\frac {\left (-\mathit {a1} x^{n +1}+\left (n +1\right ) y \right ) \lambda }{n +1}}\right )}{\left (m +1\right ) \left (m +2 n +3\right ) \left (m +n +2\right ) \mathit {a1} \lambda }, \frac {\left (-\mathit {a2} \beta x^{r +1}+\left (k +r +2\right ) x^{-k +r}\right ) \left (k +1\right )^{2} \mathit {b2} \left (-\frac {\mathit {a2} \beta x^{k +1}}{k +1}\right )^{\frac {-k -r -2}{2 k +2}} \WhittakerM \left (\frac {-k +r}{2 k +2}, \frac {2 k +r +3}{2 k +2}, -\frac {\mathit {a2} \beta x^{k +1}}{k +1}\right ) {\mathrm e}^{\frac {\mathit {a2} \beta x^{k +1}}{2 k +2}}+\left (k +r +2\right ) \left (\left (k +1\right ) \left (k +r +2\right ) \mathit {b2} x^{-k +r} \left (-\frac {\mathit {a2} \beta x^{k +1}}{k +1}\right )^{\frac {-k -r -2}{2 k +2}} \WhittakerM \left (\frac {k +r +2}{2 k +2}, \frac {2 k +r +3}{2 k +2}, -\frac {\mathit {a2} \beta x^{k +1}}{k +1}\right ) {\mathrm e}^{\frac {\mathit {a2} \beta x^{k +1}}{2 k +2}}-2 \left (r +1\right ) \left (k +\frac {r}{2}+\frac {3}{2}\right ) \mathit {a2} \,{\mathrm e}^{-\frac {\left (-\mathit {a2} x^{k +1}+\left (k +1\right ) z \right ) \beta }{k +1}}\right )}{\left (r +1\right ) \left (k +r +2\right ) \left (2 k +r +3\right ) \mathit {a2} \beta }\right ) {\mathrm e}^{\frac {c x^{s +1}}{s +1}}\]
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