Added Jan 19, 2020.
Problem Chapter 9.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c e^{\beta x} w + k x^n \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*D[w[x,y,z],y]+b*D[w[x,y,z],z]==c*Exp[beta*x]*w[x,y,z]+ k*x^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\beta x}}{\beta }} \left (\int _1^xe^{-\frac {c e^{\beta K[1]}}{\beta }} k K[1]^ndK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)=c*exp(beta*x)*w(x,y,z)+ k*x^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int k x^{n} {\mathrm e}^{-\frac {c \,{\mathrm e}^{\beta x}}{\beta }}d x +\mathit {\_F1} \left (-a x +y , -x b +z \right )\right ) {\mathrm e}^{\frac {c \,{\mathrm e}^{\beta x}}{\beta }}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a x^n w_y + b e^{\lambda x} w_z = c e^{\gamma x} w + s x^k \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*x^n*D[w[x,y,z],y]+b*Exp[lambda*x]*D[w[x,y,z],z]==c*Exp[gamma*x]*w[x,y,z]+ s*x^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c e^{\gamma x}}{\gamma }} \left (\int _1^xe^{-\frac {c e^{\gamma K[1]}}{\gamma }} s K[1]^kdK[1]+c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},z-\frac {b e^{\lambda x}}{\lambda }\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*x^n*diff(w(x,y,z),y)+ b*exp(lambda*x)*diff(w(x,y,z),z)=c*exp(gamma*x)*w(x,y,z)+ s*x^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int s x^{k} {\mathrm e}^{-\frac {c \,{\mathrm e}^{\gamma x}}{\gamma }}d x +\mathit {\_F1} \left (\frac {-a x x^{n}+\left (n +1\right ) y}{n +1}, \frac {-b \,{\mathrm e}^{\lambda x}+\lambda z}{\lambda }\right )\right ) {\mathrm e}^{\frac {c \,{\mathrm e}^{\gamma x}}{\gamma }}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + b e^{\beta x} w_y + c y^n w_z = a w + s e^{\gamma x} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ b*Exp[beta*x]*D[w[x,y,z],y]+c*y^n*D[w[x,y,z],z]==a*w[x,y,z]+ s*Exp[gamma*x]; 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)+ b*exp(beta*x)*diff(w(x,y,z),y)+ c*y^n*diff(w(x,y,z),z)=a*w(x,y,z)+ s*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (-s \,{\mathrm e}^{-\left (a -\gamma \right ) x}+\left (a -\gamma \right ) \mathit {\_F1} \left (\frac {-b \,{\mathrm e}^{\beta x}+\beta y}{\beta }, z -\left (\int _{}^{x}c \left (\frac {b \,{\mathrm e}^{\mathit {\_a} \beta }-b \,{\mathrm e}^{\beta x}+\beta y}{\beta }\right )^{n}d\mathit {\_a} \right )\right )\right ) {\mathrm e}^{a x}}{a -\gamma }\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 y+a_2 x y^k) w_y + (b_1 x+b_2 e^{\beta y+\lambda z}) w_z = c_1 w + c_2 e^{\gamma x} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*y+a2*x*y^k)*D[w[x,y,z],y]+(b1*x+b2*Exp[beta*y+lambda*z])*D[w[x,y,z],z]==c1*w[x,y,z]+ c2*Exp[gamma*x]; 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__1*y+a__2*x*y^k)*diff(w(x,y,z),y)+ (b__1*x+b__2*exp(beta*y+lambda*z))*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (-c_{2} \,{\mathrm e}^{-\left (c_{1} -\gamma \right ) x}+\left (c_{1} -\gamma \right ) \mathit {\_F1} \left (\frac {\left (\left (k -1\right ) a_{1}^{2} y +\left (\left (k -1\right ) a_{1} x -1\right ) a_{2} y^{k}\right ) y^{-k} {\mathrm e}^{\left (k -1\right ) a_{1} x}}{\left (k -1\right ) a_{1}^{2}}, \frac {-b_{2} \lambda \left (\int _{}^{x}{\mathrm e}^{\frac {\mathit {\_a}^{2} b_{1} \lambda }{2}+\beta \left (\frac {\left (-\left (\left (k -1\right ) \mathit {\_a} a_{1} -1\right ) a_{2} y^{k}+\left (\left (k -1\right ) a_{1}^{2} y +\left (\left (k -1\right ) a_{1} x -1\right ) a_{2} y^{k}\right ) {\mathrm e}^{\left (-\mathit {\_a} +x \right ) \left (k -1\right ) a_{1}}\right ) y^{-k}}{\left (k -1\right ) a_{1}^{2}}\right )^{-\frac {1}{k -1}}}d\mathit {\_a} \right )-{\mathrm e}^{\frac {\left (b_{1} x^{2}-2 z \right ) \lambda }{2}}}{\lambda }\right )\right ) {\mathrm e}^{c_{1} x}}{c_{1} -\gamma }\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 x+a_2 e^{\lambda y}) w_y + (b_1 z+b_2 e^{\beta y} z^k) w_z = c_1 w + c_2 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*x+a2*Exp[lambda*y])*D[w[x,y,z],y]+(b1*z+b2*Exp[beta*y]*z^k)*D[w[x,y,z],z]==c1*w[x,y,z]+ c2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {\text {c2}}{\text {c1}}+e^{\text {c1} x} c_1\left (-\frac {\frac {\sqrt {2 \pi } \text {a2} \sqrt {\lambda } \operatorname {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )}{\text {a1}^{3/2}}+\frac {2 e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a1}}}{2 \text {a2} \lambda ^2},(k-1) \int _1^x\text {b2} \exp \left (\frac {1}{2} \text {a1} \beta K[1]^2+\text {b1} (k-1) K[1]-\frac {\beta \left (\log (\text {a1})+\log (\text {a2})+2 \log (\lambda )+\log \left (\frac {\sqrt {\lambda } \sqrt {2 \pi } \operatorname {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )+\frac {2 \sqrt {\text {a1}} e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a2}}-\sqrt {\lambda } \sqrt {2 \pi } \operatorname {Erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } K[1]}{\sqrt {2}}\right )}{2 \text {a1}^{3/2} \lambda ^2}\right )\right )}{\lambda }\right )dK[1]+z^{1-k} e^{\text {b1} (k-1) x}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*x+a__2*exp(lambda*y))*diff(w(x,y,z),y)+ (b__1*z+b__2*exp(beta*y)*z^k)*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {c_{1} \mathit {\_F1} \left (\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {-a_{1} \lambda }\, a_{2} \erf \left (\frac {\sqrt {2}\, \sqrt {-a_{1} \lambda }\, x}{2}\right ) \sqrt {{\mathrm e}^{-\left (a_{1} x^{2}-2 y \right ) \lambda }}+2 a_{1}}{2 a_{1} \lambda \sqrt {{\mathrm e}^{-\left (a_{1} x^{2}-2 y \right ) \lambda }}}, \left (k -1\right ) b_{2} 2^{\frac {\beta }{2 \lambda }} \pi ^{-\frac {\beta }{2 \lambda }} \left (\int _{}^{x}\pi ^{\frac {\beta }{\lambda }} \left (-\frac {4 a_{1}^{3} \lambda }{\left (2 \pi a_{1} a_{2} \lambda \erf \left (\frac {\sqrt {-2 a_{1} \lambda }\, \mathit {\_a}}{2}\right )+\frac {\left (\sqrt {2}\, \sqrt {\pi }\, \sqrt {-a_{1} \lambda }\, a_{2} \erf \left (\frac {\sqrt {2}\, \sqrt {-a_{1} \lambda }\, x}{2}\right ) \sqrt {{\mathrm e}^{-\left (a_{1} x^{2}-2 y \right ) \lambda }}+2 a_{1} \right ) \sqrt {-2 \pi a_{1} \lambda }}{\sqrt {{\mathrm e}^{-\left (a_{1} x^{2}-2 y \right ) \lambda }}}\right )^{2}}\right )^{\frac {\beta }{2 \lambda }} {\mathrm e}^{\frac {\mathit {\_a}^{2} a_{1} \beta }{2}+\left (k -1\right ) \mathit {\_a} b_{1}}d\mathit {\_a} \right )+z^{-k +1} {\mathrm e}^{\left (k -1\right ) b_{1} x}\right ) {\mathrm e}^{c_{1} x}-c_{2}}{c_{1}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\mu x}+a_2 e^{\lambda y}) w_y + (b_1 e^{\nu y}+b_2 e^{\beta z}) w_z = c_1 w + c_2 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[mu*x]+a2*Exp[lambda*y])*D[w[x,y,z],y]+(b1*Exp[nu*y]+b2*Exp[beta*z])*D[w[x,y,z],z]==c1*w[x,y,z]+ c2; 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__1*exp(mu*x)+a__2*exp(lambda*y))*diff(w(x,y,z),y)+ (b__1*exp(nu*y)+b__2*exp(beta*z))*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {c_{1} \mathit {\_F1} \left (\frac {a_{2} \lambda \Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\mu x}}{\mu }\right )-\mu \,{\mathrm e}^{-\frac {\left (-a_{1} \,{\mathrm e}^{\mu x}+\mu y \right ) \lambda }{\mu }}}{\lambda \mu }, \frac {-b_{2} \beta \left (\int _{}^{x}{\mathrm e}^{b_{1} \beta \left (\int \left (\frac {-\left (-\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\mathit {\_b} \mu }}{\mu }\right )+\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\mu x}}{\mu }\right )\right ) a_{2} \lambda +\mu \,{\mathrm e}^{-\frac {\left (-a_{1} \,{\mathrm e}^{\mu x}+\mu y \right ) \lambda }{\mu }}}{\mu }\right )^{-\frac {\nu }{\lambda }} {\mathrm e}^{\frac {a_{1} \nu \,{\mathrm e}^{\mathit {\_b} \mu }}{\mu }}d \mathit {\_b} \right )}d\mathit {\_b} \right )-{\mathrm e}^{\left (b_{1} \left (\int _{}^{x}\left (\frac {-\left (-\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\mathit {\_b} \mu }}{\mu }\right )+\Ei \left (1, -\frac {a_{1} \lambda \,{\mathrm e}^{\mu x}}{\mu }\right )\right ) a_{2} \lambda +\mu \,{\mathrm e}^{-\frac {\left (-a_{1} \,{\mathrm e}^{\mu x}+\mu y \right ) \lambda }{\mu }}}{\mu }\right )^{-\frac {\nu }{\lambda }} {\mathrm e}^{\frac {a_{1} \nu \,{\mathrm e}^{\mathit {\_b} \mu }}{\mu }}d\mathit {\_b} \right )-z \right ) \beta }}{\beta }\right ) {\mathrm e}^{c_{1} x}-c_{2}}{c_{1}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\lambda _1 x} y+a_2 e^{\lambda _2 x}) w_y + (b_1 e^{\beta _1 x}z+b_2 e^{\beta _2 x}) w_z = c_1 e^{\gamma _1 x} w + c_2 e^{\gamma _2 x} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[lambda1*x]*y+a2*Exp[lambda2*x])*D[w[x,y,z],y]+(b1*Exp[beta1*x]*z+b2*Exp[beta2*x])*D[w[x,y,z],z]==c1*Exp[gamma1*x]*w[x,y,z]+ c2*Exp[gamma2*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {c1} e^{\text {gamma1} x}}{\text {gamma1}}} \left (\int _1^x\text {c2} e^{\text {gamma2} K[3]-\frac {\text {c1} e^{\text {gamma1} K[3]}}{\text {gamma1}}}dK[3]+c_1\left (y e^{-\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}}-\int _1^x\text {a2} e^{\text {lambda2} K[1]-\frac {\text {a1} e^{\text {lambda1} K[1]}}{\text {lambda1}}}dK[1],z e^{-\frac {\text {b1} e^{\text {beta1} x}}{\text {beta1}}}-\int _1^x\text {b2} e^{\text {beta2} K[2]-\frac {\text {b1} e^{\text {beta1} K[2]}}{\text {beta1}}}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*exp(lambda__1*x)*y+a__2*exp(lambda__2*x))*diff(w(x,y,z),y)+ (b__1*exp(beta__1*x)+b__2*exp(beta__2*x))*diff(w(x,y,z),z)=c__1*exp(gamma__1*x)*w(x,y,z)+ c__2*exp(gamma__2*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int c_{2} \,{\mathrm e}^{\frac {x \gamma _{1} \gamma _{2} -c_{1} \,{\mathrm e}^{x \gamma _{1}}}{\gamma _{1}}}d x +\mathit {\_F1} \left (-a_{2} \left (\int {\mathrm e}^{\frac {x \lambda _{1} \lambda _{2} -a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}d x \right )+y \,{\mathrm e}^{-\frac {a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}, \frac {-b_{1} \beta _{2} \,{\mathrm e}^{\beta _{1} x}-b_{2} \beta _{1} \,{\mathrm e}^{\beta _{2} x}+\beta _{1} \beta _{2} z}{\beta _{1} \beta _{2}}\right )\right ) {\mathrm e}^{\frac {c_{1} \,{\mathrm e}^{x \gamma _{1}}}{\gamma _{1}}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\lambda _1 x} y+a_2 e^{\lambda _2 x} y^k) w_y + (b_1 e^{\beta _1 x}z+b_2 e^{\beta _2 x} z^m) w_z = c_1 e^{\gamma _1 x} w + c_2 e^{\gamma _2 y} \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[lambda1*x]*y+a2*Exp[lambda2*x]*y^k)*D[w[x,y,z],y]+(b1*Exp[beta1*x]*z+b2*Exp[beta2*x]*z^m)*D[w[x,y,z],z]==c1*Exp[gamma1*x]*w[x,y,z]+ c2*Exp[gamma2*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {c1} e^{\text {gamma1} x}}{\text {gamma1}}} \left (\int _1^x\text {c2} \exp \left (\text {gamma2} \left (e^{-\frac {\text {a1} \left (e^{\text {lambda1} K[3]} (k-1)+e^{\text {lambda1} x}\right )}{\text {lambda1}}} y^{-k} \left (e^{\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}} (k-1) \int _1^x\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1] y^k-e^{\frac {\text {a1} e^{\text {lambda1} x}}{\text {lambda1}}} (k-1) \int _1^{K[3]}\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1] y^k+e^{\frac {\text {a1} e^{\text {lambda1} x} k}{\text {lambda1}}} y\right )\right ){}^{\frac {1}{1-k}}-\frac {\text {c1} e^{\text {gamma1} K[3]}}{\text {gamma1}}\right )dK[3]+c_1\left ((k-1) \int _1^x\text {a2} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {lambda2} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {b1} e^{\text {beta1} K[2]} (m-1)}{\text {beta1}}+\text {beta2} K[2]}dK[2]+z^{1-m} e^{\frac {\text {b1} (m-1) e^{\text {beta1} x}}{\text {beta1}}}\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*exp(lambda__1*x)*y+a__2*exp(lambda__2*x)*y^k)*diff(w(x,y,z),y)+ (b__1*exp(beta__1*x)*z+b__2*exp(beta__2*x)*z^m)*diff(w(x,y,z),z)=c__1*exp(gamma__1*x)*w(x,y,z)+ c__2*exp(gamma__2*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int _{}^{x}c_{2} \,{\mathrm e}^{\frac {\gamma _{1} \gamma _{2} \left (\left (-\left (k -1\right ) a_{2} y^{k} \left (\int {\mathrm e}^{\frac {\mathit {\_b} \lambda _{1} \lambda _{2} +\left (k -1\right ) a_{1} \,{\mathrm e}^{\mathit {\_b} \lambda _{1}}}{\lambda _{1}}}d \mathit {\_b} \right ) {\mathrm e}^{\frac {a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}+\left (k -1\right ) a_{2} y^{k} \left (\int {\mathrm e}^{\frac {x \lambda _{1} \lambda _{2} +\left (k -1\right ) a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}d x \right ) {\mathrm e}^{\frac {a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}+y \,{\mathrm e}^{\frac {a_{1} k \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}\right ) y^{-k} {\mathrm e}^{-\frac {a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\frac {a_{1} \,{\mathrm e}^{\mathit {\_b} \lambda _{1}}}{\lambda _{1}}}-c_{1} \,{\mathrm e}^{\mathit {\_b} \gamma _{1}}}{\gamma _{1}}}d\mathit {\_b} +\mathit {\_F1} \left (\left (\left (k -1\right ) a_{2} y^{k} \left (\int {\mathrm e}^{\frac {x \lambda _{1} \lambda _{2} +\left (k -1\right ) a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}d x \right ) {\mathrm e}^{\frac {a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}+y \,{\mathrm e}^{\frac {a_{1} k \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}\right ) y^{-k} {\mathrm e}^{-\frac {a_{1} \,{\mathrm e}^{x \lambda _{1}}}{\lambda _{1}}}, \left (\left (m -1\right ) b_{2} z^{m} \left (\int {\mathrm e}^{\frac {\beta _{1} \beta _{2} x +\left (m -1\right ) b_{1} \,{\mathrm e}^{\beta _{1} x}}{\beta _{1}}}d x \right ) {\mathrm e}^{\frac {b_{1} \,{\mathrm e}^{\beta _{1} x}}{\beta _{1}}}+z \,{\mathrm e}^{\frac {b_{1} m \,{\mathrm e}^{\beta _{1} x}}{\beta _{1}}}\right ) z^{-m} {\mathrm e}^{-\frac {b_{1} \,{\mathrm e}^{\beta _{1} x}}{\beta _{1}}}\right )\right ) {\mathrm e}^{\frac {c_{1} \,{\mathrm e}^{x \gamma _{1}}}{\gamma _{1}}}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + (a_1 e^{\lambda _1 x} y+a_2 e^{\lambda _2 x} y^k) w_y + (b_1 e^{\beta _1 y}z+b_2 e^{\beta _2 y} z^m) w_z = c_1 e^{\gamma _1 x} w + c_2 e^{\gamma _2 z} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ (a1*Exp[lambda1*x]*y+a2*Exp[lambda2*x]*y^k)*D[w[x,y,z],y]+(b1*Exp[beta1*y]*z+b2*Exp[beta2*y]*z^m)*D[w[x,y,z],z]==c1*Exp[gamma1*x]*w[x,y,z]+ c2*Exp[gamma2*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
$Aborted
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a__1*exp(lambda__1*x)*y+a__2*exp(lambda__2*x)*y^k)*diff(w(x,y,z),y)+ (b__1*exp(beta__1*y)*z+b__2*exp(beta__2*y)*z^m)*diff(w(x,y,z),z)=c__1*exp(gamma__1*x)*w(x,y,z)+ c__2*exp(gamma__2*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[\text {Expression too large to display}\]
____________________________________________________________________________________
Added Jan 19, 2020.
Problem Chapter 9.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 e^{\beta y} w_x + a_2 e^{sigma x} w_y + (b_1 x^n e^{\mu y}+b_2 y^m e^{\nu x+\lambda z}) w_z = c_1 w + c_2 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Exp[beta*y]*D[w[x,y,z],x]+ a2*Exp[sigma*x]*D[w[x,y,z],y]+(b1*x^n*Exp[mu*y]+b2*Exp[nu*x+lambda*z])*D[w[x,y,z],z]==c1*w[x,y,z]+ c2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a__1*exp(beta*y)*diff(w(x,y,z),x)+ a__1*exp(sigma*x)*diff(w(x,y,z),y)+ (b__1*x^n*exp(mu*y)+b__2*exp(nu*x+lambda*z))*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (c_{1} \left (\frac {\sigma \,{\mathrm e}^{\beta y}}{\beta }\right )^{\frac {c_{1}}{\left (\beta \,{\mathrm e}^{\sigma x}-\sigma \,{\mathrm e}^{\beta y}\right ) a_{1}}} \mathit {\_F1} \left (\frac {-\beta \,{\mathrm e}^{\sigma x}+\sigma \,{\mathrm e}^{\beta y}}{\beta \sigma }, \frac {-b_{2} \lambda \sigma \left (\int _{}^{x}\frac {\beta \,{\mathrm e}^{\frac {\mathit {\_b} a_{1} \beta \nu +b_{1} \lambda \sigma \left (\int \frac {\mathit {\_b}^{n} \left (-\frac {-\sigma \,{\mathrm e}^{\beta y}+\left (-{\mathrm e}^{\mathit {\_b} \sigma }+{\mathrm e}^{\sigma x}\right ) \beta }{\sigma }\right )^{\frac {\mu }{\beta }}}{{\mathrm e}^{\mathit {\_b} \sigma }-\frac {\beta \,{\mathrm e}^{\sigma x}-\sigma \,{\mathrm e}^{\beta y}}{\beta }}d \mathit {\_b} \right )}{a_{1} \beta }}}{\beta \,{\mathrm e}^{\mathit {\_b} \sigma }-\beta \,{\mathrm e}^{\sigma x}+\sigma \,{\mathrm e}^{\beta y}}d\mathit {\_b} \right )-a_{1} \beta \,{\mathrm e}^{\frac {\left (-a_{1} \beta z +b_{1} \sigma \left (\int _{}^{x}\frac {\mathit {\_b}^{n} \left (-\frac {-\sigma \,{\mathrm e}^{\beta y}+\left (-{\mathrm e}^{\mathit {\_b} \sigma }+{\mathrm e}^{\sigma x}\right ) \beta }{\sigma }\right )^{\frac {\mu }{\beta }}}{{\mathrm e}^{\mathit {\_b} \sigma }-\frac {\beta \,{\mathrm e}^{\sigma x}-\sigma \,{\mathrm e}^{\beta y}}{\beta }}d\mathit {\_b} \right )\right ) \lambda }{a_{1} \beta }}}{a_{1} \beta \lambda }\right )-c_{2} \,{\mathrm e}^{\frac {c_{1} \sigma x}{\left (\beta \,{\mathrm e}^{\sigma x}-\sigma \,{\mathrm e}^{\beta y}\right ) a_{1}}}\right ) \left ({\mathrm e}^{\sigma x}\right )^{-\frac {c_{1}}{\left (\beta \,{\mathrm e}^{\sigma x}-\sigma \,{\mathrm e}^{\beta y}\right ) a_{1}}}}{c_{1}}\]
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