7.7.5 3.1

7.7.5.1 [1617] Problem 1

problem number 1617

Added June 11, 2019.

Problem Chapter 7.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a e^{\lambda x} w_y + b e^{\beta x} w_z = c e^{\gamma x} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*x]*D[w[x,y,z],z]== c*Exp[gamma*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+ a*exp(lambda*x)*diff(w(x,y,z),y)+b*exp(beta*x)*diff(w(x,y,z),z)=  c*exp(gamma*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {c \,{\mathrm e}^{\gamma x}+\gamma \textit {\_F1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+\lambda y}{\lambda }, \frac {-b \,{\mathrm e}^{\beta x}+\beta z}{\beta }\right )}{\gamma }\]

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7.7.5.2 [1618] Problem 2

problem number 1618

Added June 11, 2019.

Problem Chapter 7.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a e^{\lambda x} w_y + b e^{\beta y} w_z = c e^{\gamma y} + s e^{\mu z} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] +b*Exp[beta*y]*D[w[x,y,z],z]== c*Exp[gamma*y]+s*Exp[mu*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+ a*exp(lambda*x)*diff(w(x,y,z),y)+b*exp(beta*y)*diff(w(x,y,z),z)=  c*exp(gamma*y)+s*exp(mu*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\left (c \,{\mathrm e}^{-\frac {\left (-a \,{\mathrm e}^{\textit {\_a} \lambda }+a \,{\mathrm e}^{\lambda x}-\lambda y \right ) \gamma }{\lambda }}+s \,{\mathrm e}^{\frac {\left (\left (-\Ei \left (1, -\frac {a \beta \,{\mathrm e}^{\textit {\_a} \lambda }}{\lambda }\right )+\Ei \left (1, -\frac {a \beta \,{\mathrm e}^{\lambda x}}{\lambda }\right )\right ) b \,{\mathrm e}^{-\frac {\left (a \,{\mathrm e}^{\lambda x}-\lambda y \right ) \beta }{\lambda }}+\lambda z \right ) \mu }{\lambda }}\right )d \textit {\_a} +\textit {\_F1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+\lambda y}{\lambda }, \frac {b \Ei \left (1, -\frac {a \beta \,{\mathrm e}^{\lambda x}}{\lambda }\right ) {\mathrm e}^{-\frac {\left (a \,{\mathrm e}^{\lambda x}-\lambda y \right ) \beta }{\lambda }}+\lambda z}{\lambda }\right )\]

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7.7.5.3 [1619] Problem 3

problem number 1619

Added June 11, 2019.

Problem Chapter 7.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a e^{\lambda y} w_y + b e^{\beta y} w_z = c e^{\gamma x} + s e^{\mu z} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Exp[lambda*y]*D[w[x, y,z], y] +b*Exp[beta*y]*D[w[x,y,z],z]== c*Exp[gamma*x]+s*Exp[mu*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+ a*exp(lambda*y)*diff(w(x,y,z),y)+b*exp(beta*y)*diff(w(x,y,z),z)=  c*exp(gamma*x)+s*exp(mu*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\left (c \,{\mathrm e}^{\textit {\_a} \gamma }+s \,{\mathrm e}^{\frac {\left (-b \left ({\mathrm e}^{\lambda y}\right )^{\frac {\beta }{\lambda }} {\mathrm e}^{-\lambda y}+\left (\beta -\lambda \right ) a z +\left (\left (-\textit {\_a} +x \right ) a \lambda +{\mathrm e}^{-\lambda y}\right ) b \left (\frac {1}{\left (-\textit {\_a} +x \right ) a \lambda +{\mathrm e}^{-\lambda y}}\right )^{\frac {\beta }{\lambda }}\right ) \mu }{\left (\beta -\lambda \right ) a}}\right )d \textit {\_a} +\textit {\_F1} \left (\frac {-a \lambda x -{\mathrm e}^{-\lambda y}}{a \lambda }, \frac {-b \left ({\mathrm e}^{\lambda y}\right )^{\frac {\beta }{\lambda }} {\mathrm e}^{-\lambda y}+\left (\beta -\lambda \right ) a z}{\left (\beta -\lambda \right ) a}\right )\]

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7.7.5.4 [1620] Problem 4

problem number 1620

Added June 11, 2019.

Problem Chapter 7.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + (A_1 e^{\alpha _1 x} +B_1 e^{\nu _1 x+\lambda y} ) w_y + (A_2 e^{\alpha _2 x} +B_2 e^{\nu _2 x+\beta y} ) w_z = k e^{\gamma z} \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (A1*Exp[alpha1*x] +B1*Exp[nu1*x+lambda*y] )*D[w[x, y,z], y] +(A2*Exp[alpha2*x] +B2*Exp[nu2*x+beta*y] )*D[w[x,y,z],z]== k*Exp[gamma*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(alpha1*x) +B1*exp(nu1*x+lambda*y) )*diff(w(x,y,z),y)+(A2*exp(alpha2*x) +B2*exp(nu2*x+beta*y) )*diff(w(x,y,z),z)= k*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}k \,{\mathrm e}^{\left (z +\int \left (\mathit {B2} \left (-\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\textit {\_f} \alpha 1}+\textit {\_f} \alpha 1 \nu 1}{\alpha 1}}d \textit {\_f} \right )+\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\alpha 1 x}}{\alpha 1}+\nu 1 x}d x \right )+{\mathrm e}^{\frac {\left (\mathit {A1} \,{\mathrm e}^{\alpha 1 x}-\alpha 1 y \right ) \lambda }{\alpha 1}}\right )^{-\frac {\beta }{\lambda }} {\mathrm e}^{\frac {\mathit {A1} \beta \,{\mathrm e}^{\textit {\_f} \alpha 1}+\textit {\_f} \alpha 1 \nu 2}{\alpha 1}}+\mathit {A2} \,{\mathrm e}^{\textit {\_f} \alpha 2}\right )d \textit {\_f} -\left (\int _{}^{x}\left (\mathit {B2} \left (-\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\textit {\_b} \alpha 1}+\textit {\_b} \alpha 1 \nu 1}{\alpha 1}}d \textit {\_b} \right )+\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\alpha 1 x}}{\alpha 1}+\nu 1 x}d x \right )+{\mathrm e}^{\frac {\left (\mathit {A1} \,{\mathrm e}^{\alpha 1 x}-\alpha 1 y \right ) \lambda }{\alpha 1}}\right )^{-\frac {\beta }{\lambda }} {\mathrm e}^{\frac {\mathit {A1} \beta \,{\mathrm e}^{\textit {\_b} \alpha 1}+\textit {\_b} \alpha 1 \nu 2}{\alpha 1}}+\mathit {A2} \,{\mathrm e}^{\textit {\_b} \alpha 2}\right )d \textit {\_b} \right )\right ) \gamma }d \textit {\_f} +\textit {\_F1} \left (\frac {-\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\alpha 1 x}}{\alpha 1}+\nu 1 x}d x \right )-{\mathrm e}^{\frac {\left (\mathit {A1} \,{\mathrm e}^{\alpha 1 x}-\alpha 1 y \right ) \lambda }{\alpha 1}}}{\lambda }, z -\left (\int _{}^{x}\left (\mathit {B2} \left (-\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\textit {\_b} \alpha 1}+\textit {\_b} \alpha 1 \nu 1}{\alpha 1}}d \textit {\_b} \right )+\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\alpha 1 x}}{\alpha 1}+\nu 1 x}d x \right )+{\mathrm e}^{\frac {\left (\mathit {A1} \,{\mathrm e}^{\alpha 1 x}-\alpha 1 y \right ) \lambda }{\alpha 1}}\right )^{-\frac {\beta }{\lambda }} {\mathrm e}^{\frac {\mathit {A1} \beta \,{\mathrm e}^{\textit {\_b} \alpha 1}+\textit {\_b} \alpha 1 \nu 2}{\alpha 1}}+\mathit {A2} \,{\mathrm e}^{\textit {\_b} \alpha 2}\right )d \textit {\_b} \right )\right )\]

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7.7.5.5 [1621] Problem 5

problem number 1621

Added June 11, 2019.

Problem Chapter 7.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a e^{\alpha x} w_x + b e^{\beta y} w_y + c e^{\gamma z} w_z = k e^{\lambda x} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[alpha*x]*D[w[x, y,z], x] + b*Exp[beta*y]*D[w[x, y,z], y] +c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde := a*exp(alpha*x)*diff(w(x,y,z),x)+  b*exp(beta*y)*diff(w(x,y,z),y)+c*exp(gamma*z)*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'));
 

\[w \left (x , y , z\right ) = \frac {\left (-\alpha +\lambda \right ) a \textit {\_F1} \left (-\frac {\left (a \alpha \,{\mathrm e}^{\alpha x}-b \beta \,{\mathrm e}^{\beta y}\right ) {\mathrm e}^{-\alpha x -\beta y}}{\alpha b \beta }, \frac {\left (-a \alpha \,{\mathrm e}^{\alpha x}+c \gamma \,{\mathrm e}^{\gamma z}\right ) {\mathrm e}^{-\alpha x -\gamma z}}{\alpha c \gamma }\right )+k \,{\mathrm e}^{-\left (\alpha -\lambda \right ) x}}{\left (-\alpha +\lambda \right ) a}\]

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7.7.5.6 [1622] Problem 6

problem number 1622

Added June 11, 2019.

Problem Chapter 7.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a e^{\beta y} w_x + b e^{\alpha x} w_y + c e^{\gamma z} w_z = k e^{\lambda x} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[beta*x]*D[w[x, y,z], x] + b*Exp[alpha*x]*D[w[x, y,z], y] +c*Exp[gamma*z]*D[w[x,y,z],z]== k*Exp[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde := a*exp(beta*y)*diff(w(x,y,z),x)+  b*exp(alpha*x)*diff(w(x,y,z),y)+c*exp(gamma*z)*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'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\frac {\alpha k \,{\mathrm e}^{\textit {\_a} \lambda }}{a \alpha \,{\mathrm e}^{\beta y}-\left (-{\mathrm e}^{\textit {\_a} \alpha }+{\mathrm e}^{\alpha x}\right ) b \beta }d \textit {\_a} +\textit {\_F1} \left (\frac {a \alpha \,{\mathrm e}^{\beta y}-b \beta \,{\mathrm e}^{\alpha x}}{\alpha b \beta }, -\frac {\left (\alpha c \gamma x -c \gamma \ln \left (\frac {a \alpha \,{\mathrm e}^{\beta y}}{b \beta }\right )+\left (a \alpha \,{\mathrm e}^{\beta y}-b \beta \,{\mathrm e}^{\alpha x}\right ) {\mathrm e}^{-\gamma z}\right ) b \beta }{\left (a \alpha \,{\mathrm e}^{\beta y}-b \beta \,{\mathrm e}^{\alpha x}\right ) \alpha c \gamma }\right )\]

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7.7.5.7 [1623] Problem 7

problem number 1623

Added June 11, 2019.

Problem Chapter 7.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ (a_1+a_2 e^{\alpha x}) w_x + (b_1+b_2 e^{\beta y}) w_y + (c_1+c_2 e^{\gamma z}) w_z = k_1 + k_2 e^{\alpha x} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + (b1+b2*Exp[beta*y])*D[w[x, y,z], y] +(c1+c2*Exp[gamma*z])*D[w[x,y,z],z]== k1+k2*Exp[alpha*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \frac {(\text {a1} \text {k2}-\text {a2} \text {k1}) \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )+\text {a2} \alpha \text {k1} x}{\text {a1} \text {a2} \alpha }+c_1\left (\frac {\log \left (\frac {e^{\beta y} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {b1} \beta }{\text {a1} \alpha }}}{\text {b1}+\text {b2} e^{\beta y}}\right )}{\text {b1} \beta }-\frac {x}{\text {a1}},\frac {\log \left (\frac {e^{\gamma z} \left (\text {a1}+\text {a2} e^{\alpha x}\right )^{\frac {\text {c1} \gamma }{\text {a1} \alpha }}}{\text {c1}+\text {c2} e^{\gamma z}}\right )}{\text {c1} \gamma }-\frac {x}{\text {a1}}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := (a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+ (b1+b2*exp(beta*y))*diff(w(x,y,z),y)+(c1+c2*exp(gamma*z))*diff(w(x,y,z),z)= k1+k2*exp(alpha*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 {\mathit {a1} \mathit {a2} \alpha \textit {\_F1} \left (\frac {-\mathit {b1} \beta x +\mathit {a1} \ln \left (\frac {\left (-\mathit {b1} +\RootOf \left (\mathit {a1} \alpha \beta y -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {b1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {b1} \beta }{\mathit {a1} \alpha }}}{\mathit {b2}}\right )+\mathit {b1} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )\right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {b1} \beta }{\mathit {a1} \alpha }}}{\mathit {b2} \RootOf \left (\mathit {a1} \alpha \beta y -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {b1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {b1} \beta }{\mathit {a1} \alpha }}}{\mathit {b2}}\right )+\mathit {b1} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}\right )}{\mathit {a1} \mathit {b1} \beta }, \frac {-\mathit {c1} \gamma x +\mathit {a1} \ln \left (\frac {\left (-\mathit {c1} +\RootOf \left (\mathit {a1} \alpha \gamma z -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {c1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {c1} \gamma }{\mathit {a1} \alpha }}}{\mathit {c2}}\right )+\mathit {c1} \gamma \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )\right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {c1} \gamma }{\mathit {a1} \alpha }}}{\mathit {c2} \RootOf \left (\mathit {a1} \alpha \gamma z -\mathit {a1} \alpha \ln \left (\frac {\left (\textit {\_Z} -\mathit {c1} \right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {c1} \gamma }{\mathit {a1} \alpha }}}{\mathit {c2}}\right )+\mathit {c1} \gamma \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}\right )}{\mathit {a1} \mathit {c1} \gamma }\right )+\mathit {a1} \mathit {k2} \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )-\mathit {a2} \mathit {k1} \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )+\mathit {a2} \mathit {k1} \ln \left ({\mathrm e}^{\alpha x}\right )}{\mathit {a1} \mathit {a2} \alpha }\]

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7.7.5.8 [1624] Problem 8

problem number 1624

Added June 11, 2019.

Problem Chapter 7.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ e^{\beta y}(a_1+a_2 e^{\alpha x}) w_x + e^{\alpha x} (b_1+b_2 e^{\beta y}) w_y + c e^{\beta y+\gamma z} w_z = k_3 e^{\beta y} (k_1 + k_2 e^{\alpha x}) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = Exp[beta*y]*(a1+a2*Exp[alpha*x])*D[w[x, y,z], x] + Exp[alpha*x]*(b1+b2*Exp[beta*y])*D[w[x, y,z], y] +c*Exp[beta*y+gamma*z]*D[w[x,y,z],z]== k3*Exp[beta*y]*(k1+k2*Exp[alpha*x]); 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \frac {\text {k3} \left ((\text {a1} \text {k2}-\text {a2} \text {k1}) \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )+\text {a2} \alpha \text {k1} x\right )}{\text {a1} \text {a2} \alpha }+c_1\left (\frac {c \log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a1} \alpha }-\frac {c x}{\text {a1}}-\frac {e^{-\gamma z}}{\gamma },\frac {\log \left (\text {b1}+\text {b2} e^{\beta y}\right )}{\text {b2} \beta }-\frac {\log \left (\text {a1}+\text {a2} e^{\alpha x}\right )}{\text {a2} \alpha }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := exp(beta*y)*(a1+a2*exp(alpha*x))*diff(w(x,y,z),x)+  exp(alpha*x)*(b1+b2*exp(beta*y))*diff(w(x,y,z),y)+c*exp(beta*y+gamma*z)*diff(w(x,y,z),z)= k3*exp(beta*y)*(k1+k2*exp(alpha*x)); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {\left (\mathit {k1} \mathit {k3} x +\mathit {a1} \textit {\_F1} \left (\frac {\ln \left (\frac {\mathit {b1} \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {b2} \beta }{\mathit {a2} \alpha }} \RootOf \left (\mathit {a2} \alpha \beta y -\mathit {a2} \alpha \ln \left (\frac {\mathit {b1} \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {b2} \beta }{\mathit {a2} \alpha }}}{\textit {\_Z} -\mathit {b2}}\right )-\mathit {b2} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}{-\mathit {b2} +\RootOf \left (\mathit {a2} \alpha \beta y -\mathit {a2} \alpha \ln \left (\frac {\mathit {b1} \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{-\frac {\mathit {b2} \beta }{\mathit {a2} \alpha }}}{\textit {\_Z} -\mathit {b2}}\right )-\mathit {b2} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )}\right )}{\mathit {b2} \beta }, \frac {c \gamma \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )-\left (c \gamma x +\mathit {a1} \,{\mathrm e}^{-\gamma z}\right ) \alpha }{\mathit {a1} \alpha c \gamma }\right )\right ) \mathit {a2} \alpha +\left (\mathit {a1} \mathit {k2} -\mathit {a2} \mathit {k1} \right ) \mathit {k3} \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )}{\mathit {a1} \mathit {a2} \alpha }\]

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