Added June 11, 2019.
Problem Chapter 7.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a x^n w_y + b x^m w_z = c e^{\lambda x} y+ k e^{\beta x} z+ s e^{\gamma x} \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right )-\frac {a c x^n (-\lambda x)^{-n} \text {Gamma}(n+2,-\lambda x)}{\lambda ^2 (n+1)}-\frac {b k x^m (-\beta x)^{-m} \text {Gamma}(m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {c e^{\lambda x} \left (-a x^{n+1}+n y+y\right )}{\lambda (n+1)}-\frac {b k e^{\beta x} x^{m+1}}{\beta m+\beta }+\frac {k z e^{\beta x}}{\beta }+\frac {s e^{\gamma x}}{\gamma }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*x^n*diff(w(x,y,z),y)+b*x^m*diff(w(x,y,z),z)=c*exp(lambda*x)*y+k*exp(beta*x)*z+s*exp(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{\gamma \,{\lambda }^{2}{\beta }^{2} \left ( 1+n \right ) \left ( m+1 \right ) } \left ( -b{x}^{m}k \left ( \Gamma \left ( m,-\beta \,x \right ) -\Gamma \left ( m \right ) \right ) \left ( -\beta \right ) ^{m}m{\lambda }^{2} \left ( 1+n \right ) \left ( m+1 \right ) \left ( -\beta \right ) ^{-m}\gamma \, \left ( -\beta \,x \right ) ^{-m}- \left ( \left ( -x\lambda +n+1 \right ) {{\rm e}^{x\lambda }}+n \left ( -x\lambda \right ) ^{-n} \left ( \Gamma \left ( n,-x\lambda \right ) -\Gamma \left ( n \right ) \right ) \left ( 1+n \right ) \right ) {x}^{n} \left ( -\lambda \right ) ^{n}ca \left ( m+1 \right ) \gamma \,{\beta }^{2} \left ( -\lambda \right ) ^{-n}+\lambda \, \left ( \gamma \,{x}^{m}\lambda \, \left ( -\beta \right ) ^{m}b{{\rm e}^{\beta \,x}}k \left ( 1+n \right ) \left ( \beta \,x-m-1 \right ) \left ( -\beta \right ) ^{-m}+ \left ( -a\beta \,\gamma \,c \left ( m+1 \right ) \left ( {{\rm e}^{x\lambda }}-1 \right ) {x}^{1+n}+ \left ( -\gamma \,\lambda \,bk \left ( {{\rm e}^{\beta \,x}}-1 \right ) {x}^{m+1}+ \left ( \beta \,\gamma \,cy{{\rm e}^{x\lambda }}+z\gamma \,\lambda \,{{\rm e}^{\beta \,x}}k+\beta \,\lambda \,s{{\rm e}^{x\gamma }}+ \left ( \left ( {\it \_F1} \left ( {\frac {-{x}^{1+n}a+y \left ( 1+n \right ) }{1+n}},{\frac {-{x}^{m+1}b+z \left ( m+1 \right ) }{m+1}} \right ) \gamma -s \right ) \beta -z\gamma \,k \right ) \lambda -\beta \,\gamma \,cy \right ) \left ( m+1 \right ) \right ) \left ( 1+n \right ) \right ) \beta \right ) \right ) }\]
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a e^{\lambda x} w_y + b x^m w_z = c x^n y+ k e^{\beta x} z+ s e^{\gamma x} \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {-b x^{m+1}+m z+z}{m+1}\right )+\frac {a c x^n (-\lambda x)^{-n} \text {Gamma}(n+1,-\lambda x)}{\lambda ^2}-\frac {b k x^m (-\beta x)^{-m} \text {Gamma}(m+2,-\beta x)}{\beta ^2 (m+1)}-\frac {a c e^{\lambda x} x^{n+1}}{\lambda n+\lambda }+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {c y x^{n+1}}{n+1}+\frac {s e^{\gamma x}}{\gamma }\right \}\right \}\]
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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {{{\rm e}^{\beta \,{\it \_a}}}{{\it \_a}}^{m+1}bk\lambda -\lambda \,{x}^{m+1}{{\rm e}^{\beta \,{\it \_a}}}bk- \left ( -z\lambda \,{{\rm e}^{\beta \,{\it \_a}}}k-{{\rm e}^{{\it \_a}\,\lambda }}{{\it \_a}}^{n}ac-s{{\rm e}^{{\it \_a}\,\gamma }}\lambda +{{\it \_a}}^{n} \left ( a{{\rm e}^{x\lambda }}-\lambda \,y \right ) c \right ) \left ( m+1 \right ) }{\lambda \, \left ( m+1 \right ) }}{d{\it \_a}}+{\it \_F1} \left ( {\frac {-a{{\rm e}^{x\lambda }}+\lambda \,y}{\lambda }},{\frac {-{x}^{m+1}b+z \left ( m+1 \right ) }{m+1}} \right ) \]
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a e^{\lambda x} w_y + b y w_z = k e^{\beta x} z+ s e^{\gamma x} \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {a b e^{\lambda x} (\lambda x-1)}{\lambda ^2}-b x y+z\right )+\frac {a b k e^{x (\beta +\lambda )}}{\beta ^2 (\beta +\lambda )}-\frac {b k y e^{\beta x}}{\beta ^2}+\frac {k z e^{\beta x}}{\beta }+\frac {s e^{\gamma x}}{\gamma }\right \}\right \}\]
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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{{\lambda }^{2}{\beta }^{2} \left ( \beta +\lambda \right ) \gamma } \left ( {\lambda }^{2}{\beta }^{2} \left ( \beta +\lambda \right ) \gamma \,{\it \_F1} \left ( {\frac {-a{{\rm e}^{x\lambda }}+\lambda \,y}{\lambda }},{\frac {ab \left ( x\lambda -1 \right ) {{\rm e}^{x\lambda }}-{\lambda }^{2} \left ( bxy-z \right ) }{{\lambda }^{2}}} \right ) +kab{{\rm e}^{x \left ( \beta +\lambda \right ) }}{\beta }^{2}\gamma + \left ( \beta +\lambda \right ) \left ( \left ( ab \left ( -\beta +\lambda \right ) {{\rm e}^{x\lambda }}-{\lambda }^{2} \left ( by-\beta \,z \right ) \right ) \gamma \,k{{\rm e}^{\beta \,x}}+{{\rm e}^{x\gamma }}{\beta }^{2}{\lambda }^{2}s \right ) \right ) }\]
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a y^n w_y + b z^m w_z = c e^{\lambda x}+ k e^{\beta y}+ s e^{\gamma z} \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-a x-\frac {\left (\frac {1}{y}\right )^{n-1}}{n-1},-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right )+\frac {k \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {n}{n-1}} \left (-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )^n \text {Gamma}\left (1-n,-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \text {Gamma}\left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right \}\right \}\]
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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!c{{\rm e}^{{\it \_a}\,\lambda }}+k{{\rm e}^{\beta \, \left ( {\frac {a \left ( x-{\it \_a} \right ) \left ( n-1 \right ) {y}^{n}+y}{{y}^{n}}} \right ) ^{- \left ( n-1 \right ) ^{-1}}}}+s{{\rm e}^{\gamma \, \left ( {\frac {b \left ( x-{\it \_a} \right ) \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) ^{- \left ( m-1 \right ) ^{-1}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ax \left ( n-1 \right ) {y}^{n}+y}{{y}^{n}}},{\frac {bx \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) \]
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a e^{\beta y} w_y + b z^m w_z = c e^{\lambda x}+ k y^n + s e^{\gamma z} \]
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]== x*Exp[lambda*x]+k*y^n+s*Exp[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a \beta x+e^{-\beta y}}{\beta },-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right )+\frac {k \log ^{n-1}\left (e^{-\beta y}\right ) \left (-\frac {\log \left (e^{-\beta y}\right )}{\beta }\right )^{n+1} \left (-\log ^2\left (e^{-\beta y}\right )\right )^{-n} \text {Gamma}\left (n+1,-\log \left (e^{-\beta y}\right )\right )}{a}+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \text {Gamma}\left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }-\frac {e^{\lambda x}}{\lambda ^2}+\frac {x e^{\lambda x}}{\lambda }\right \}\right \}\]
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)=x*exp(lambda*x)+k*y^n+s*exp(gamma*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\it \_a}\,{{\rm e}^{{\it \_a}\,\lambda }}+k \left ( {\frac {\ln \left ( \left ( {{\rm e}^{-\beta \,y}}+\beta \,a \left ( x-{\it \_a} \right ) \right ) ^{-1} \right ) }{\beta }} \right ) ^{n}+s{{\rm e}^{\gamma \, \left ( {\frac {b \left ( x-{\it \_a} \right ) \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) ^{- \left ( m-1 \right ) ^{-1}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {-a\beta \,x-{{\rm e}^{-\beta \,y}}}{a\beta }},{\frac {bx \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) \]
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( y^2+b y+a e^{\alpha x}(y-b)-b^2 \right ) w_y + \left ( z^2+c(x z-1) e^{\beta x} \right ) w_z = k e^{\gamma x} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + ( y^2+b*y+a*Exp[alpha*x]*(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[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)+ ( y^2+b*y+a*exp(alpha*x)*(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(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( y^2+a e^{\alpha x}(x+1) \right ) w_y + \left ( c e^{\beta x} z^2+b e^{-\beta x} \right ) w_z = k e^{\lambda x} \]
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]; 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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( a e^{\alpha x} y^2+b e^{-\alpha x} \right ) w_y + \left ( d e^{\beta x} z^2 +c e^{\gamma x}(\gamma -c d e^{(\beta +\gamma )x})\right ) w_z = k e^{\lambda x} \]
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]; 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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[\text {Expression too large to display}\]
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( a_1 e^{\lambda _1 x} y+ b_1 e^{\beta _1 x} y^k \right ) w_y + \left ( a_2 e^{\lambda _2 x} z +b_2 e^{\beta _2 x}z^m \right ) w_z = c x^s \]
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; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {c x^{s+1}}{s+1}+c_1\left ((k-1) \int _1^x\text {b1} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {beta1} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {a2} e^{\text {lambda2} K[2]} (m-1)}{\text {lambda2}}+\text {beta2} K[2]}dK[2]+z^{1-m} e^{\frac {\text {a2} (m-1) e^{\text {lambda2} x}}{\text {lambda2}}}\right )\right \}\right \}\]
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; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{s+1} \left ( \left ( s+1 \right ) {\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}}{\lambda 1}}}}{\it b1}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}} \left ( k-1 \right ) +\beta 1\,x\lambda 1}{\lambda 1}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}k}{\lambda 1}}}}y \right ) \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}}{\lambda 1}}}} \right ) ^{-1}},{\frac {1}{{z}^{m}} \left ( {{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}}}{\lambda 2}}}}{\it b2}\,{z}^{m} \left ( m-1 \right ) \int \!{{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}} \left ( m-1 \right ) +\beta 2\,x\lambda 2}{\lambda 2}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}}m}{\lambda 2}}}}z \right ) \left ( {{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}}}{\lambda 2}}}} \right ) ^{-1}} \right ) +{x}^{s+1}c \right ) }\]
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( a_1 e^{\beta _1 x} y+ b_1 e^{\gamma _1 x} y^k \right ) w_y + \left ( a_2 e^{\beta _2 x} z +b_2 e^{\gamma _1 x+\lambda z} \right ) w_z = c x^s \]
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]*z +b2*Exp[gamma1*x+lambda*z])*D[w[x,y,z],z]== c*x^s; 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)*z +b2*exp(gamma1*x+lambda*z))*diff(w(x,y,z),z)=c*x^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added June 11, 2019.
Problem Chapter 7.3.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^n+b_1 x^m e^{\lambda y} ) w_y + \left ( a_2 x^k+b_2 x^L e^{\beta z}\right ) w_z = c x^s \]
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^L*Exp[beta*z])*D[w[x,y,z],z]== c*x^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {c x^{s+1}}{s+1}+c_1\left (\frac {\text {b2} \beta x^{L+1} \left (-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )^{-\frac {L+1}{k+1}} \text {Gamma}\left (\frac {L+1}{k+1},-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\beta \left (-\text {a2} x^{k+1}+k z+z\right )}{k+1}}}{\text {a2} \text {b2} \beta ^2 (k+1) (k-L)},\frac {(n+1) e^{-\frac {\lambda \left (-\text {a1} x^{n+1}+n y+y\right )}{n+1}}-\text {b1} \lambda x^{m+1} \left (-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \text {Gamma}\left (\frac {m+1}{n+1},-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )}{\text {a1} \text {b1} \lambda ^2 (n+1) (m-n)}\right )\right \}\right \}\]
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^L*exp(beta*z))*diff(w(x,y,z),z)=c*x^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{s+1} \left ( \left ( s+1 \right ) {\it \_F1} \left ( {\frac {1}{ \left ( m+1 \right ) \left ( n+m+2 \right ) \left ( 2\,n+m+3 \right ) {\it a1}\,\lambda } \left ( {\it b1}\, \left ( -{\frac {{x}^{1+n}{\it a1}\,\lambda }{1+n}} \right ) ^{{\frac {-2-n-m}{2+2\,n}}} \left ( \left ( n+m+2 \right ) {x}^{m-n}-{\it a1}\,\lambda \,{x}^{m+1} \right ) {{\rm e}^{{\frac {{x}^{1+n}{\it a1}\,\lambda }{2+2\,n}}}} \left ( 1+n \right ) ^{2} \WhittakerM \left ( {\frac {m-n}{2+2\,n}},{\frac {2\,n+m+3}{2+2\,n}},-{\frac {{x}^{1+n}{\it a1}\,\lambda }{1+n}} \right ) - \left ( -{x}^{m-n}{{\rm e}^{{\frac {{x}^{1+n}{\it a1}\,\lambda }{2+2\,n}}}} \left ( -{\frac {{x}^{1+n}{\it a1}\,\lambda }{1+n}} \right ) ^{{\frac {-2-n-m}{2+2\,n}}}{\it b1}\, \left ( 1+n \right ) \left ( n+m+2 \right ) \WhittakerM \left ( {\frac {n+m+2}{2+2\,n}},{\frac {2\,n+m+3}{2+2\,n}},-{\frac {{x}^{1+n}{\it a1}\,\lambda }{1+n}} \right ) +{{\rm e}^{-{\frac { \left ( -{x}^{1+n}{\it a1}+y \left ( 1+n \right ) \right ) \lambda }{1+n}}}}{\it a1}\, \left ( m+1 \right ) \left ( 2\,n+m+3 \right ) \right ) \left ( n+m+2 \right ) \right ) },{\frac {1}{ \left ( L+1 \right ) \left ( L+k+2 \right ) \left ( L+2\,k+3 \right ) {\it a2}\,\beta } \left ( {\it b2}\, \left ( -{\frac {{x}^{1+k}{\it a2}\,\beta }{1+k}} \right ) ^{{\frac {-L-k-2}{2+2\,k}}} \left ( \left ( L+k+2 \right ) {x}^{L-k}-{\it a2}\,\beta \,{x}^{L+1} \right ) {{\rm e}^{{\frac {{x}^{1+k}{\it a2}\,\beta }{2+2\,k}}}} \left ( 1+k \right ) ^{2} \WhittakerM \left ( {\frac {L-k}{2+2\,k}},{\frac {L+2\,k+3}{2+2\,k}},-{\frac {{x}^{1+k}{\it a2}\,\beta }{1+k}} \right ) - \left ( -{x}^{L-k}{{\rm e}^{{\frac {{x}^{1+k}{\it a2}\,\beta }{2+2\,k}}}} \left ( -{\frac {{x}^{1+k}{\it a2}\,\beta }{1+k}} \right ) ^{{\frac {-L-k-2}{2+2\,k}}}{\it b2}\, \left ( 1+k \right ) \left ( L+k+2 \right ) \WhittakerM \left ( {\frac {L+k+2}{2+2\,k}},{\frac {L+2\,k+3}{2+2\,k}},-{\frac {{x}^{1+k}{\it a2}\,\beta }{1+k}} \right ) +{{\rm e}^{-{\frac { \left ( -{x}^{1+k}{\it a2}+z \left ( 1+k \right ) \right ) \beta }{1+k}}}}{\it a2}\, \left ( L+1 \right ) \left ( L+2\,k+3 \right ) \right ) \left ( L+k+2 \right ) \right ) } \right ) +{x}^{s+1}c \right ) }\]
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