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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (-\frac {a c x^n (-\lambda x)^{-n} \text {Gamma}(n+2,-\lambda x)}{\lambda ^2 (n+1)}-\frac {b k x^m (-\beta x)^{-m} \text {Gamma}(m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {c e^{\lambda x} \left (-a x^{n+1}+n y+y\right )}{\lambda (n+1)}+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-a{x}^{n+1}+y \left ( n+1 \right ) }{n+1}},{\frac {-{x}^{m+1}c+z \left ( m+1 \right ) }{m+1}} \right ) {{\rm e}^{{\frac {a{\beta }^{2}{x}^{n}cn \left ( n+1 \right ) \left ( m+1 \right ) \left ( \Gamma \left ( n \right ) -\Gamma \left ( n,-\lambda \,x \right ) \right ) \left ( -\lambda \,x \right ) ^{-n}+{x}^{m}{\lambda }^{2}ckm \left ( n+1 \right ) \left ( m+1 \right ) \left ( \Gamma \left ( m \right ) -\Gamma \left ( m,-\beta \,x \right ) \right ) \left ( -\beta \,x \right ) ^{-m}+8\,s{{\rm e}^{x/8}} \left ( m+1 \right ) \left ( n+1 \right ) {\beta }^{2}{\lambda }^{2}+\beta \,{\lambda }^{2}ck \left ( n+1 \right ) {x}^{m+1}- \left ( -\lambda \,{\beta }^{2}{x}^{n+1}ac+ \left ( c{\beta }^{2} \left ( {x}^{n}a-y\lambda \right ) {{\rm e}^{\lambda \,x}}-{\lambda }^{2}k \left ( \beta \,z-c{x}^{m} \right ) {{\rm e}^{\beta \,x}}+ \left ( y\lambda \,c+8\,{\lambda }^{2}s \right ) {\beta }^{2}+\beta \,{\lambda }^{2}kz \right ) \left ( n+1 \right ) \right ) \left ( m+1 \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) {\beta }^{2}{\lambda }^{2}}}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (\frac {a c x^n (-\lambda x)^{-n} \text {Gamma}(n+1,-\lambda x)}{\lambda ^2}-\frac {b k x^m (-\beta x)^{-m} \text {Gamma}(m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {c x^{n+1} \left (\lambda y-a e^{\lambda x}\right )}{\lambda (n+1)}+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-a{{\rm e}^{\lambda \,x}}+y\lambda }{\lambda }},{\frac {-b{x}^{m+1}+z \left ( m+1 \right ) }{m+1}} \right ) {{\rm e}^{\int ^{x}\!{\frac {s{{\rm e}^{{\it \_a}/8}}\lambda \, \left ( m+1 \right ) +{{\it \_a}}^{m+1}{{\rm e}^{\beta \,{\it \_a}}}bk\lambda -\lambda \,{x}^{m+1}{{\rm e}^{\beta \,{\it \_a}}}bk- \left ( m+1 \right ) \left ( -z\lambda \,{{\rm e}^{\beta \,{\it \_a}}}k+c{{\it \_a}}^{n} \left ( a{{\rm e}^{\lambda \,x}}-a{{\rm e}^{{\it \_a}\,\lambda }}-y\lambda \right ) \right ) }{\lambda \, \left ( m+1 \right ) }}{d{\it \_a}}}}\]
____________________________________________________________________________________
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-a{{\rm e}^{\lambda \,x}}+y\lambda }{\lambda }},{\frac {ab \left ( \lambda \,x-1 \right ) {{\rm e}^{\lambda \,x}}-{\lambda }^{2} \left ( bxy-z \right ) }{{\lambda }^{2}}} \right ) {{\rm e}^{{\frac {kab{{\rm e}^{x \left ( \beta +\lambda \right ) }}{\beta }^{2}- \left ( \beta +\lambda \right ) \left ( -8\,{{\rm e}^{x/8}}{\beta }^{2}{\lambda }^{2}s+ \left ( ab \left ( \beta -\lambda \right ) {{\rm e}^{\lambda \,x}}+{\lambda }^{2} \left ( by-\beta \,z \right ) \right ) k{{\rm e}^{\beta \,x}} \right ) }{{\beta }^{2}{\lambda }^{2} \left ( \beta +\lambda \right ) }}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-a x-\frac {\left (\frac {1}{y}\right )^{n-1}}{n-1},-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right ) \exp \left (\frac {k \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {n}{n-1}} \left (-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )^n \text {Gamma}\left (1-n,-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \text {Gamma}\left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ax \left ( n-1 \right ) {y}^{n}+y}{{y}^{n}}},{\frac {xb \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) {{\rm e}^{\int ^{x}\!{{\rm e}^{{\it \_a}\,\lambda }}c+k{{\rm e}^{\beta \, \left ( {\frac {a \left ( x-{\it \_a} \right ) \left ( n-1 \right ) {y}^{n}+y}{{y}^{n}}} \right ) ^{- \left ( n-1 \right ) ^{-1}}}}+s{{\rm e}^{1/8\, \left ( {\frac {b \left ( x-{\it \_a} \right ) \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) ^{- \left ( m-1 \right ) ^{-1}}}}{d{\it \_a}}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a \beta x+e^{-\beta y}}{\beta },-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right ) \exp \left (-\frac {k \left (-\log \left (e^{-\beta y}\right )\right )^{-n} \left (-\frac {\log \left (e^{-\beta y}\right )}{\beta }\right )^n \text {Gamma}\left (n+1,-\log \left (e^{-\beta y}\right )\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \text {Gamma}\left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-\beta \,xa-{{\rm e}^{-\beta \,y}}}{a\beta }},{\frac {xb \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) {{\rm e}^{\int ^{x}\!{{\rm e}^{{\it \_a}\,\lambda }}c+s{{\rm e}^{1/8\, \left ( {\frac {b \left ( x-{\it \_a} \right ) \left ( m-1 \right ) {z}^{m}+z}{{z}^{m}}} \right ) ^{- \left ( m-1 \right ) ^{-1}}}}+k \left ( {\frac {\ln \left ( \left ( {{\rm e}^{-\beta \,y}}+a\beta \, \left ( x-{\it \_a} \right ) \right ) ^{-1} \right ) }{\beta }} \right ) ^{n}{d{\it \_a}}}}\]
____________________________________________________________________________________
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'));
\[{\it PDESolStruc} \left ( w \left ( x,y,z \right ) ={\it \_F1} \left ( x \right ) {\it \_C1}\,{{\rm e}^{{\it \_c}_{{2}}\int \! \left ( a \left ( b-y \right ) {{\rm e}^{\alpha \,y}}+{b}^{2}-by-{y}^{2} \right ) ^{-1}\,{\rm d}y}}{\it \_F3} \left ( z \right ) ,[ \left \{ c \left ( {\frac {\rm d}{{\rm d}z}}{\it \_F3} \left ( z \right ) \right ) {\it \_F1} \left ( x \right ) \left ( zx-1 \right ) {{\rm e}^{\beta \,x}}-{\it \_F1} \left ( x \right ) {\it \_F3} \left ( z \right ) {{\rm e}^{\lambda \,x}}k+{\it \_F1} \left ( x \right ) \left ( {\frac {\rm d}{{\rm d}z}}{\it \_F3} \left ( z \right ) \right ) {z}^{2}+{\it \_F3} \left ( z \right ) \left ( -{\it \_c}_{{2}}{\it \_F1} \left ( x \right ) +{\frac {\rm d}{{\rm d}x}}{\it \_F1} \left ( x \right ) \right ) =0 \right \} ] \right ) \]
____________________________________________________________________________________
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'));
\[\text {Expression too large to display}\]
____________________________________________________________________________________
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}}{\lambda 1}}}}{\it b1}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}} \left ( k-1 \right ) +\beta 1\,x\lambda 1}{\lambda 1}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}k}{\lambda 1}}}}y \right ) \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\lambda 1\,x}}}{\lambda 1}}}} \right ) ^{-1}},{\frac {1}{{z}^{m}} \left ( {{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}}}{\lambda 2}}}}{\it b2}\,{z}^{m} \left ( m-1 \right ) \int \!{{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}} \left ( m-1 \right ) +\beta 2\,x\lambda 2}{\lambda 2}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}}m}{\lambda 2}}}}z \right ) \left ( {{\rm e}^{{\frac {{\it a2}\,{{\rm e}^{\lambda 2\,x}}}{\lambda 2}}}} \right ) ^{-1}} \right ) {{\rm e}^{{\frac {{x}^{s+1}c}{s+1}}}}\]
____________________________________________________________________________________
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}}}{\beta 1}}}}{\it b1}\,{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}} \left ( k-1 \right ) +\gamma 1\,x\beta 1}{\beta 1}}}}\,{\rm d}x+{{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}}k}{\beta 1}}}}y \right ) \left ( {{\rm e}^{{\frac {{\it a1}\,{{\rm e}^{\beta 1\,x}}}{\beta 1}}}} \right ) ^{-1}},{\frac {1}{\lambda 2} \left ( -{\it b2}\,\int \!{{\rm e}^{\gamma 2\,x+{\frac {\lambda 2\,{\it a2}\,{{\rm e}^{\beta 2\,x}}}{\beta 2}}}}\,{\rm d}x\lambda 2-{{\rm e}^{-{\frac {\lambda 2\, \left ( \beta 2\,z-{\it a2}\,{{\rm e}^{\beta 2\,x}} \right ) }{\beta 2}}}} \right ) } \right ) {{\rm e}^{{\frac {{x}^{s+1}c}{s+1}}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left (\frac {(n+1) e^{-\frac {\lambda \left (-\text {a1} x^{n+1}+n y+y\right )}{n+1}}-\text {b1} \lambda x^{m+1} \left (-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \text {Gamma}\left (\frac {m+1}{n+1},-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )}{\text {a1} \text {b1} \lambda ^2 (n+1) (m-n)},\frac {\text {b2} \beta x^{r+1} \left (-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )^{-\frac {r+1}{k+1}} \text {Gamma}\left (\frac {r+1}{k+1},-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\beta \left (-\text {a2} x^{k+1}+k z+z\right )}{k+1}}}{\text {a2} \text {b2} \beta ^2 (k+1) (k-r)}\right )\right \}\right \}\]
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 ) ={\it \_F1} \left ( {\frac {1}{ \left ( m+1 \right ) \left ( m+n+2 \right ) \left ( m+2\,n+3 \right ) {\it a1}\,\lambda } \left ( \left ( n+1 \right ) ^{2} \left ( \left ( m+n+2 \right ) {x}^{-n+m}-{\it a1}\,\lambda \,{x}^{m+1} \right ) {{\rm e}^{{\frac {{x}^{n+1}\lambda \,{\it a1}}{2\,n+2}}}}{\it b1}\, \left ( -{\frac {{x}^{n+1}\lambda \,{\it a1}}{n+1}} \right ) ^{{\frac {-n-2-m}{2\,n+2}}} \WhittakerM \left ( {\frac {-n+m}{2\,n+2}},{\frac {m+2\,n+3}{2\,n+2}},-{\frac {{x}^{n+1}\lambda \,{\it a1}}{n+1}} \right ) - \left ( -{{\rm e}^{{\frac {{x}^{n+1}\lambda \,{\it a1}}{2\,n+2}}}} \left ( -{\frac {{x}^{n+1}\lambda \,{\it a1}}{n+1}} \right ) ^{{\frac {-n-2-m}{2\,n+2}}}{\it b1}\,{x}^{-n+m} \left ( n+1 \right ) \left ( m+n+2 \right ) \WhittakerM \left ( {\frac {m+n+2}{2\,n+2}},{\frac {m+2\,n+3}{2\,n+2}},-{\frac {{x}^{n+1}\lambda \,{\it a1}}{n+1}} \right ) +{{\rm e}^{-{\frac { \left ( -{x}^{n+1}{\it a1}+y \left ( n+1 \right ) \right ) \lambda }{n+1}}}}{\it a1}\, \left ( m+1 \right ) \left ( m+2\,n+3 \right ) \right ) \left ( m+n+2 \right ) \right ) },{\frac {1}{\beta \,{\it a2}\, \left ( r+1 \right ) \left ( k+r+2 \right ) \left ( 2\,k+r+3 \right ) } \left ( \left ( \left ( k+r+2 \right ) {x}^{-k+r}-{x}^{r+1}{\it a2}\,\beta \right ) {{\rm e}^{{\frac {{x}^{k+1}\beta \,{\it a2}}{2\,k+2}}}} \left ( k+1 \right ) ^{2}{\it b2}\, \left ( -{\frac {{x}^{k+1}\beta \,{\it a2}}{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 {{x}^{k+1}\beta \,{\it a2}}{k+1}} \right ) + \left ( {{\rm e}^{{\frac {{x}^{k+1}\beta \,{\it a2}}{2\,k+2}}}} \left ( -{\frac {{x}^{k+1}\beta \,{\it a2}}{k+1}} \right ) ^{{\frac {-k-r-2}{2\,k+2}}}{\it b2}\,{x}^{-k+r} \left ( k+1 \right ) \left ( k+r+2 \right ) \WhittakerM \left ( {\frac {k+r+2}{2\,k+2}},{\frac {2\,k+r+3}{2\,k+2}},-{\frac {{x}^{k+1}\beta \,{\it a2}}{k+1}} \right ) -2\, \left ( k+r/2+3/2 \right ) {{\rm e}^{-{\frac { \left ( -{x}^{k+1}{\it a2}+z \left ( k+1 \right ) \right ) \beta }{k+1}}}} \left ( r+1 \right ) {\it a2} \right ) \left ( k+r+2 \right ) \right ) } \right ) {{\rm e}^{{\frac {{x}^{s+1}c}{s+1}}}}\]
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