6.7.6 3.2

6.7.6.1 [1623] Problem 1

problem number 1623

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right )-\frac {a c x^n (-\lambda x)^{-n} \Gamma (n+2,-\lambda x)}{\lambda ^2 (n+1)}+\frac {c e^{\lambda x} \left (-a x^{n+1}+n y+y\right )}{\lambda (n+1)}-\frac {b k x^m (-\beta x)^{-m} \Gamma (m+2,-\beta x)}{\beta ^2 (m+1)}-\frac {b k e^{\beta x} x^{m+1}}{\beta m+\beta }+\frac {k z e^{\beta x}}{\beta }+\frac {s e^{\gamma x}}{\gamma }\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \frac {-\lambda ^{2} \gamma k \left (n +1\right ) b \left (m +1\right ) \left (m \Gamma \left (m , -\beta x \right )-\Gamma \left (m +1\right )\right ) x^{m} \left (-\beta x \right )^{-m}+\beta \gamma \,\lambda ^{2} b k \left (n +1\right ) x^{m +1}+\left (a \,x^{n +1} c \,\beta ^{2} \gamma \lambda +\left (-\beta ^{2} \gamma c \left (a \,x^{n}-\lambda y \right ) {\mathrm e}^{\lambda x}+{\mathrm e}^{\gamma x} \beta ^{2} \lambda ^{2} s +\gamma \,\lambda ^{2} k \left (z \beta -b \,x^{m}\right ) {\mathrm e}^{\beta x}+\beta \left (\left (-\lambda x \right )^{-n} a \beta \gamma \,x^{n} \Gamma \left (n +1\right ) c -\Gamma \left (n , -\lambda x \right ) \left (-\lambda x \right )^{-n} a \beta \gamma \,x^{n} c n +\lambda \left (\left (\left (f_{1} \left (\frac {-a \,x^{n +1}+y \left (n +1\right )}{n +1}, \frac {-b \,x^{m +1}+z \left (m +1\right )}{m +1}\right ) \gamma -s \right ) \beta -\gamma k z \right ) \lambda -\beta c \gamma y \right )\right )\right ) \left (n +1\right )\right ) \left (m +1\right )}{\beta ^{2} \lambda ^{2} \gamma \left (n +1\right ) \left (m +1\right )}\]

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6.7.6.2 [1624] Problem 2

problem number 1624

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {-b x^{m+1}+m z+z}{m+1}\right )+\frac {a c x^n (-\lambda x)^{-n} \Gamma (n+1,-\lambda x)}{\lambda ^2}-\frac {a c e^{\lambda x} x^{n+1}}{\lambda n+\lambda }-\frac {b k x^m (-\beta x)^{-m} \Gamma (m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {c y x^{n+1}}{n+1}+\frac {s e^{\gamma x}}{\gamma }\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \frac {\int _{}^{x}\left ({\mathrm e}^{\beta \textit {\_a}} \textit {\_a}^{m +1} b k \lambda -{\mathrm e}^{\beta \textit {\_a}} x^{m +1} b k \lambda -\left (-{\mathrm e}^{\beta \textit {\_a}} k \lambda z -\textit {\_a}^{n} {\mathrm e}^{\lambda \textit {\_a}} a c -s \,{\mathrm e}^{\gamma \textit {\_a}} \lambda +\textit {\_a}^{n} \left (-y \lambda +a \,{\mathrm e}^{\lambda x}\right ) c \right ) \left (m +1\right )\right )d \textit {\_a} +\lambda \left (m +1\right ) f_{1} \left (\frac {y \lambda -a \,{\mathrm e}^{\lambda x}}{\lambda }, \frac {-b \,x^{m +1}+z \left (m +1\right )}{m +1}\right )}{\lambda \left (m +1\right )}\]

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6.7.6.3 [1625] Problem 3

problem number 1625

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]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\left (e^{\gamma K[2]} s+\frac {e^{\beta K[2]} k \left (a b e^{\lambda K[2]} K[2]+b \left (\lambda y-a e^{\lambda x}\right ) K[2]-\lambda \left (b x y-z+\int _1^x-a b e^{\lambda K[1]} K[1]dK[1]\right )+\lambda \int _1^{K[2]}-a b e^{\lambda K[1]} K[1]dK[1]\right )}{\lambda }\right )dK[2]+c_1\left (y-\frac {a e^{\lambda x}}{\lambda },-\int _1^x-a b e^{\lambda K[1]} K[1]dK[1]-b x y+z\right )\right \}\right \}\]

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); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {\beta ^{2} \left (\beta +\lambda \right ) \gamma f_{1} \left (\frac {y \lambda -a \,{\mathrm e}^{\lambda x}}{\lambda }, \frac {a b \left (\lambda x -1\right ) {\mathrm e}^{\lambda x}-\lambda ^{2} \left (b x y -z \right )}{\lambda ^{2}}\right )+{\mathrm e}^{x \left (\beta +\lambda \right )} a b \gamma k -\left (\beta +\lambda \right ) \left (k \gamma \left (b y -\beta z \right ) {\mathrm e}^{\beta x}-{\mathrm e}^{\gamma x} s \,\beta ^{2}\right )}{\beta ^{2} \left (\beta +\lambda \right ) \gamma }\]

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6.7.6.4 [1626] Problem 4

problem number 1626

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-a x-\frac {\left (\frac {1}{y}\right )^{n-1}}{n-1},-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right )+\frac {k \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {n}{n-1}} \left (-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )^n \Gamma \left (1-n,-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \Gamma \left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\left (c \,{\mathrm e}^{\lambda \textit {\_a}}+k \,{\mathrm e}^{\beta \left (y^{1-n}+\left (\left (1-n \right ) \textit {\_a} +x n -x \right ) a \right )^{-\frac {1}{n -1}}}+s \,{\mathrm e}^{\gamma \left (z^{1-m}+\left (\left (1-m \right ) \textit {\_a} +x m -x \right ) b \right )^{-\frac {1}{m -1}}}\right )d \textit {\_a} +f_{1} \left (y^{1-n}+\left (n -1\right ) x a , z^{1-m}+\left (m -1\right ) x b \right )\]

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6.7.6.5 [1627] Problem 5

problem number 1627

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]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\left (k \left (-\frac {\log \left (a \beta (x-K[1])+e^{-\beta y}\right )}{\beta }\right )^n+\exp \left (\gamma \left (\left (\frac {1}{z}\right )^{m-1}+b (m-1) (x-K[1])\right )^{\frac {1}{1-m}}\right ) s+e^{\lambda K[1]} K[1]\right )dK[1]+c_1\left (-\frac {a \beta x+e^{-\beta y}}{\beta },-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right )\right \}\right \}\]

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)=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'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\left (\textit {\_a} \,{\mathrm e}^{\lambda \textit {\_a}}+k {\left (\frac {\ln \left (\frac {1}{{\mathrm e}^{-\beta y}+\left (x -\textit {\_a} \right ) a \beta }\right )}{\beta }\right )}^{n}+s \,{\mathrm e}^{\gamma \left (z^{1-m}+\left (\left (1-m \right ) \textit {\_a} +x m -x \right ) b \right )^{-\frac {1}{m -1}}}\right )d \textit {\_a} +f_{1} \left (\frac {-{\mathrm e}^{-\beta y}-a \beta x}{a \beta }, z^{1-m}+\left (m -1\right ) x b \right )\]

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6.7.6.6 [1628] Problem 6

problem number 1628

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=()

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6.7.6.7 [1629] Problem 7

problem number 1629

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=()

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6.7.6.8 [1630] Problem 8

problem number 1630

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'));
 

\[\text {Expression too large to display}\]

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6.7.6.9 [1631] Problem 9

problem number 1631

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'));
 

\[w \left (x , y , z\right ) = \frac {\left (s +1\right ) f_{1} \left (\operatorname {b1} \left (k -1\right ) \int {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\lambda \operatorname {1} x} \left (k -1\right )+\beta \operatorname {1} x \lambda \operatorname {1} }{\lambda \operatorname {1}}}d x +y^{1-k} {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\lambda \operatorname {1} x} \left (k -1\right )}{\lambda \operatorname {1}}}, \operatorname {b2} \left (m -1\right ) \int {\mathrm e}^{\frac {\operatorname {a2} \,{\mathrm e}^{\lambda \operatorname {2} x} \left (m -1\right )+\beta \operatorname {2} x \lambda \operatorname {2} }{\lambda \operatorname {2}}}d x +z^{1-m} {\mathrm e}^{\frac {\operatorname {a2} \,{\mathrm e}^{\lambda \operatorname {2} x} \left (m -1\right )}{\lambda \operatorname {2}}}\right )+c \,x^{s +1}}{s +1}\]

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6.7.6.10 [1632] Problem 10

problem number 1632

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=()

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6.7.6.11 [1633] Problem 11

problem number 1633

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]];
 

\[\left \{\left \{w(x,y,z)\to \frac {c x^{s+1}}{s+1}+c_1\left (\frac {\text {b2} \beta x^{L+1} \left (-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )^{-\frac {L+1}{k+1}} \Gamma \left (\frac {L+1}{k+1},-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\beta \left (-\text {a2} x^{k+1}+k z+z\right )}{k+1}}}{\text {a2} \text {b2} \beta ^2 (k+1) (k-L)},\frac {(n+1) e^{-\frac {\lambda \left (-\text {a1} x^{n+1}+n y+y\right )}{n+1}}-\text {b1} \lambda x^{m+1} \left (-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \Gamma \left (\frac {m+1}{n+1},-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )}{\text {a1} \text {b1} \lambda ^2 (n+1) (m-n)}\right )\right \}\right \}\]

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 {\left (s +1\right ) f_{1} \left (-\frac {\left (-\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )^{\frac {-m -n -2}{2 n +2}} x^{-n} \left (-{\mathrm e}^{\frac {x \lambda \operatorname {a1} \,x^{n}}{2 n +2}} \operatorname {b1} \,x^{m} \left (n +1\right )^{2} \left (-x \lambda \operatorname {a1} \,x^{n}+m +n +2\right ) \operatorname {WhittakerM}\left (\frac {m -n}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )+\left (-{\mathrm e}^{\frac {x \lambda \operatorname {a1} \,x^{n}}{2 n +2}} \operatorname {b1} \,x^{m} \left (n +1\right ) \left (m +n +2\right ) \operatorname {WhittakerM}\left (\frac {m +n +2}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )+{\mathrm e}^{-\frac {\left (-\operatorname {a1} x \,x^{n}+y \left (n +1\right )\right ) \lambda }{n +1}} \operatorname {a1} \,x^{n} \left (-\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )^{\frac {m +n +2}{2 n +2}} \left (m +1\right ) \left (m +2 n +3\right )\right ) \left (m +n +2\right )\right )}{\operatorname {a1} \lambda \left (m +1\right ) \left (m +2 n +3\right ) \left (m +n +2\right )}, -\frac {\left (-\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )^{\frac {-L -k -2}{2 k +2}} x^{-k} \left (-{\mathrm e}^{\frac {x^{k} \operatorname {a2} \beta x}{2 k +2}} \operatorname {b2} \,x^{L} \left (k +1\right )^{2} \left (-x^{k} \operatorname {a2} \beta x +L +k +2\right ) \operatorname {WhittakerM}\left (\frac {L -k}{2 k +2}, \frac {L +2 k +3}{2 k +2}, -\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )+\left (L +k +2\right ) \left (-{\mathrm e}^{\frac {x^{k} \operatorname {a2} \beta x}{2 k +2}} \operatorname {b2} \,x^{L} \left (k +1\right ) \left (L +k +2\right ) \operatorname {WhittakerM}\left (\frac {L +k +2}{2 k +2}, \frac {L +2 k +3}{2 k +2}, -\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )+{\mathrm e}^{-\frac {\left (-x \operatorname {a2} \,x^{k}+z \left (k +1\right )\right ) \beta }{k +1}} \operatorname {a2} \,x^{k} \left (-\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )^{\frac {L +k +2}{2 k +2}} \left (L +1\right ) \left (L +2 k +3\right )\right )\right )}{\operatorname {a2} \beta \left (L +1\right ) \left (L +2 k +3\right ) \left (L +k +2\right )}\right )+x c \,x^{s}}{s +1}\]

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