6.9.6 3.2

6.9.6.1 [1961] Problem 1

problem number 1961

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.2 [1962] Problem 2

problem number 1962

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.3 [1963] Problem 3

problem number 1963

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.4 [1964] Problem 4

problem number 1964

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'));
 

\[w \left (x , y , z\right ) = \frac {{\mathrm e}^{c_{1} x} \left (c_{1} -\gamma \right ) f_{1} \left (\frac {{\mathrm e}^{a_{1} \left (k -1\right ) x} \left (\left (k -1\right ) a_{1}^{2} y^{1-k}+\left (-1+a_{1} \left (k -1\right ) x \right ) a_{2} \right )}{\left (k -1\right ) a_{1}^{2}}, \frac {-b_{2} \int _{}^{x}{\mathrm e}^{\frac {\lambda b_{1} \textit {\_a}^{2}}{2}+\beta {\left (\frac {\left (\left (k -1\right ) a_{1}^{2} y^{1-k}+\left (-1+a_{1} \left (k -1\right ) x \right ) a_{2} \right ) {\mathrm e}^{a_{1} \left (k -1\right ) \left (-\textit {\_a} +x \right )}-\left (-1+a_{1} \textit {\_a} \left (k -1\right )\right ) a_{2}}{\left (k -1\right ) a_{1}^{2}}\right )}^{-\frac {1}{k -1}}}d \textit {\_a} \lambda -{\mathrm e}^{\frac {\lambda \left (b_{1} x^{2}-2 z \right )}{2}}}{\lambda }\right )-c_{2} {\mathrm e}^{\gamma x}}{c_{1} -\gamma }\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.5 [1965] Problem 5

problem number 1965

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]];
 

\[\left \{\left \{w(x,y,z)\to -\frac {\text {c2}}{\text {c1}}+e^{\text {c1} x} c_1\left (-\frac {\frac {\sqrt {2 \pi } \text {a2} \sqrt {\lambda } \text {erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )}{\text {a1}^{3/2}}+\frac {2 e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a1}}}{2 \text {a2} \lambda ^2},(k-1) \int _1^x\text {b2} \exp \left (\frac {1}{2} \text {a1} \beta K[1]^2+\text {b1} (k-1) K[1]-\frac {\beta \left (\log (\text {a1})+\log (\text {a2})+2 \log (\lambda )+\log \left (\frac {\sqrt {\lambda } \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } x}{\sqrt {2}}\right )+\frac {2 \sqrt {\text {a1}} e^{\frac {1}{2} \text {a1} \lambda x^2-\lambda y}}{\text {a2}}-\sqrt {\lambda } \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {\text {a1}} \sqrt {\lambda } K[1]}{\sqrt {2}}\right )}{2 \text {a1}^{3/2} \lambda ^2}\right )\right )}{\lambda }\right )dK[1]+z^{1-k} e^{\text {b1} (k-1) x}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \frac {{\mathrm e}^{c_{1} x} f_{1} \left (\frac {\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-\lambda a_{1}}\, x}{2}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\lambda a_{1}}\, a_{2} \sqrt {{\mathrm e}^{-\lambda \left (a_{1} x^{2}-2 y \right )}}+2 a_{1}}{2 \sqrt {{\mathrm e}^{-\lambda \left (a_{1} x^{2}-2 y \right )}}\, \lambda a_{1}}, \pi ^{-\frac {\beta }{2 \lambda }} \left (b_{2} \pi ^{\frac {\beta }{\lambda }} 2^{\frac {\beta }{2 \lambda }} \left (k -1\right ) \int _{}^{x}{\mathrm e}^{\textit {\_a} \left (k -1\right ) b_{1} +\frac {\beta a_{1} \textit {\_a}^{2}}{2}} \left (-\frac {a_{1} {\mathrm e}^{-\lambda \left (a_{1} x^{2}-2 y \right )} \lambda }{\left (\pi a_{2} \lambda \left (\operatorname {erf}\left (\frac {\sqrt {-2 \lambda a_{1}}\, x}{2}\right )-\operatorname {erf}\left (\frac {\sqrt {-2 \lambda a_{1}}\, \textit {\_a}}{2}\right )\right ) \sqrt {{\mathrm e}^{-\lambda \left (a_{1} x^{2}-2 y \right )}}-\sqrt {-2 \lambda a_{1} \pi }\right )^{2}}\right )^{\frac {\beta }{2 \lambda }}d \textit {\_a} +z^{1-k} \pi ^{\frac {\beta }{2 \lambda }} {\mathrm e}^{b_{1} \left (k -1\right ) x}\right )\right ) c_{1} -c_{2}}{c_{1}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.6 [1966] Problem 6

problem number 1966

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'));
 

\[w \left (x , y , z\right ) = \frac {{\mathrm e}^{c_{1} x} f_{1} \left (\frac {a_{2} \operatorname {Ei}_{1}\left (-\frac {\lambda a_{1} {\mathrm e}^{\mu x}}{\mu }\right ) \lambda -{\mathrm e}^{\frac {\lambda \left (a_{1} {\mathrm e}^{\mu x}-\mu y \right )}{\mu }} \mu }{\mu \lambda }, \frac {-\int _{}^{x}{\mathrm e}^{b_{1} \beta \int {\mathrm e}^{\frac {\nu a_{1} {\mathrm e}^{\mu \textit {\_a}}}{\mu }} \left (\frac {{\mathrm e}^{\frac {\lambda \left (a_{1} {\mathrm e}^{\mu x}-\mu y \right )}{\mu }} \mu -a_{2} \lambda \left (\operatorname {Ei}_{1}\left (-\frac {\lambda a_{1} {\mathrm e}^{\mu x}}{\mu }\right )-\operatorname {Ei}_{1}\left (-\frac {\lambda a_{1} {\mathrm e}^{\mu \textit {\_a}}}{\mu }\right )\right )}{\mu }\right )^{-\frac {\nu }{\lambda }}d \textit {\_a}}d \textit {\_a} b_{2} \beta -{\mathrm e}^{\beta \left (\int _{}^{x}{\mathrm e}^{\frac {\nu a_{1} {\mathrm e}^{\mu \textit {\_a}}}{\mu }} \left (\frac {{\mathrm e}^{\frac {\lambda \left (a_{1} {\mathrm e}^{\mu x}-\mu y \right )}{\mu }} \mu -a_{2} \lambda \left (\operatorname {Ei}_{1}\left (-\frac {\lambda a_{1} {\mathrm e}^{\mu x}}{\mu }\right )-\operatorname {Ei}_{1}\left (-\frac {\lambda a_{1} {\mathrm e}^{\mu \textit {\_a}}}{\mu }\right )\right )}{\mu }\right )^{-\frac {\nu }{\lambda }}d \textit {\_a} b_{1} -z \right )}}{\beta }\right ) c_{1} -c_{2}}{c_{1}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.7 [1967] Problem 7

problem number 1967

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'));
 

\[w \left (x , y , z\right ) = \left (c_{2} \int {\mathrm e}^{\frac {\gamma _{2} x \gamma _{1} -{\mathrm e}^{\gamma _{1} x} c_{1}}{\gamma _{1}}}d x +f_{1} \left ({\mathrm e}^{-\frac {{\mathrm e}^{\lambda _{1} x} a_{1}}{\lambda _{1}}} y -a_{2} \int {\mathrm e}^{\frac {\lambda _{2} x \lambda _{1} -{\mathrm e}^{\lambda _{1} x} a_{1}}{\lambda _{1}}}d x , \frac {z \beta _{1} \beta _{2} -b_{1} {\mathrm e}^{\beta _{1} x} \beta _{2} -b_{2} {\mathrm e}^{\beta _{2} x} \beta _{1}}{\beta _{1} \beta _{2}}\right )\right ) {\mathrm e}^{\frac {{\mathrm e}^{\gamma _{1} x} c_{1}}{\gamma _{1}}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.8 [1968] Problem 8

problem number 1968

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'));
 

\[w \left (x , y , z\right ) = \left (c_{2} \int _{}^{x}{\mathrm e}^{\frac {\gamma _{2} \left (-a_{2} \left (k -1\right ) \int {\mathrm e}^{\frac {a_{1} \left (k -1\right ) {\mathrm e}^{\lambda _{1} \textit {\_b}}+\textit {\_b} \lambda _{1} \lambda _{2}}{\lambda _{1}}}d \textit {\_b} +a_{2} \left (k -1\right ) \int {\mathrm e}^{\frac {a_{1} {\mathrm e}^{\lambda _{1} x} \left (k -1\right )+\lambda _{2} x \lambda _{1}}{\lambda _{1}}}d x +y^{1-k} {\mathrm e}^{\frac {a_{1} {\mathrm e}^{\lambda _{1} x} \left (k -1\right )}{\lambda _{1}}}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\frac {a_{1} {\mathrm e}^{\lambda _{1} \textit {\_b}}}{\lambda _{1}}} \gamma _{1} -{\mathrm e}^{\gamma _{1} \textit {\_b}} c_{1}}{\gamma _{1}}}d \textit {\_b} +f_{1} \left (a_{2} \left (k -1\right ) \int {\mathrm e}^{\frac {a_{1} {\mathrm e}^{\lambda _{1} x} \left (k -1\right )+\lambda _{2} x \lambda _{1}}{\lambda _{1}}}d x +y^{1-k} {\mathrm e}^{\frac {a_{1} {\mathrm e}^{\lambda _{1} x} \left (k -1\right )}{\lambda _{1}}}, b_{2} \left (m -1\right ) \int {\mathrm e}^{\frac {b_{1} {\mathrm e}^{\beta _{1} x} \left (m -1\right )+\beta _{2} x \beta _{1}}{\beta _{1}}}d x +z^{1-m} {\mathrm e}^{\frac {b_{1} {\mathrm e}^{\beta _{1} x} \left (m -1\right )}{\beta _{1}}}\right )\right ) {\mathrm e}^{\frac {{\mathrm e}^{\gamma _{1} x} c_{1}}{\gamma _{1}}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.9 [1969] Problem 9

problem number 1969

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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.9.6.10 [1970] Problem 10

problem number 1970

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'));
 

\[w \left (x , y , z\right ) = \frac {\left (\left (\frac {{\mathrm e}^{\beta y} \sigma }{\beta }\right )^{\frac {c_{1}}{a_{1} \left (-{\mathrm e}^{\beta y} \sigma +{\mathrm e}^{\sigma x} \beta \right )}} f_{1} \left (\frac {{\mathrm e}^{\beta y}}{\beta }-\frac {{\mathrm e}^{\sigma x}}{\sigma }, \frac {\int _{}^{x}\frac {{\mathrm e}^{-\frac {\int \frac {\left (\frac {-{\mathrm e}^{\sigma x} \beta +{\mathrm e}^{\beta y} \sigma +{\mathrm e}^{\sigma \textit {\_b}} \beta }{\sigma }\right )^{\frac {\mu }{\beta }}}{{\mathrm e}^{\sigma x} \beta -{\mathrm e}^{\beta y} \sigma -{\mathrm e}^{\sigma \textit {\_b}} \beta }d \textit {\_b} \textit {\_a}^{n} b_{1} \lambda \sigma }{a_{1}}+\nu \textit {\_b}}}{{\mathrm e}^{\sigma x} \beta -{\mathrm e}^{\beta y} \sigma -{\mathrm e}^{\sigma \textit {\_b}} \beta }d \textit {\_b} b_{2} \sigma }{a_{1}}-\frac {{\mathrm e}^{-\lambda \left (\frac {\int _{}^{x}\frac {\left (\frac {-{\mathrm e}^{\sigma x} \beta +{\mathrm e}^{\beta y} \sigma +{\mathrm e}^{\sigma \textit {\_b}} \beta }{\sigma }\right )^{\frac {\mu }{\beta }}}{{\mathrm e}^{\sigma x} \beta -{\mathrm e}^{\beta y} \sigma -{\mathrm e}^{\sigma \textit {\_b}} \beta }d \textit {\_b} \textit {\_a}^{n} b_{1} \sigma }{a_{1}}+z \right )}}{\lambda }\right ) c_{1} -{\mathrm e}^{\frac {c_{1} \sigma x}{a_{1} \left (-{\mathrm e}^{\beta y} \sigma +{\mathrm e}^{\sigma x} \beta \right )}} c_{2} \right ) \left ({\mathrm e}^{\sigma x}\right )^{-\frac {c_{1}}{a_{1} \left (-{\mathrm e}^{\beta y} \sigma +{\mathrm e}^{\sigma x} \beta \right )}}}{c_{1}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________