6.9.5 3.1

6.9.5.1 [1954] Problem 1

problem number 1954

Added Jan 18, 2020.

Problem Chapter 9.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Exp[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

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

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

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6.9.5.2 [1955] Problem 2

problem number 1955

Added Jan 18, 2020.

Problem Chapter 9.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.9.5.3 [1956] Problem 3

problem number 1956

Added Jan 18, 2020.

Problem Chapter 9.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+ b*Exp[beta*x]*D[w[x,y,z],y]+c*Exp[lambda*y]*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]];
 

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

Maple ✓

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

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

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6.9.5.4 [1957] Problem 4

problem number 1957

Added Jan 18, 2020.

Problem Chapter 9.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.9.5.5 [1958] Problem 5

problem number 1958

Added Jan 18, 2020.

Problem Chapter 9.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a_1 e^{\sigma x}+ a_2 e^{\lambda y} ) w_y + (b_1 e^{\mu y}+ b_2 e^{\beta z} ) w_z = c_1 w +c_2 e^{\nu x} \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+ (a1*Exp[sigma*x]+ a2*Exp[lambda*y] )*D[w[x,y,z],y]+(b1*Exp[mu*y]+ b2*Exp[beta*z])*D[w[x,y,z],z]==c1*w[x,y,z]+ c2*Exp[nu*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*exp(sigma*x)+ a__2*exp(lambda*y) )*diff(w(x,y,z),y)+ (b__1*exp(mu*y)+ b__2*exp(beta*z))*diff(w(x,y,z),z)=c__1*w(x,y,z)+ c__2*exp(nu*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.9.5.6 [1959] Problem 6

problem number 1959

Added Jan 18, 2020.

Problem Chapter 9.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ b_1 e^{\lambda _1 x} w_x + b_2 e^{\lambda _2 y} w_y + b_3 e^{\lambda _3 z} w_z = a w +c_1 e^{\beta _1 x}+c_2 e^{\beta _2 y}+c_3 e^{\beta _3 z} \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  b1*Exp[lambda1*x]*D[w[x,y,z],x]+ b2*Exp[lambda2*y]*D[w[x,y,z],y]+b3*Exp[lambda3*z]*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Exp[beta1*x]+c2*Exp[beta2*y]+c3*Exp[beta3*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to e^{-\frac {a e^{-\text {lambda1} x}}{\text {b1} \text {lambda1}}} \left (\int _1^x\frac {e^{\frac {a e^{-\text {lambda1} K[1]}}{\text {b1} \text {lambda1}}-\text {lambda1} K[1]} \left (\text {c2} \left (\frac {\text {b2} \left (-e^{-\text {lambda1} x}+e^{-\text {lambda1} K[1]}\right ) \text {lambda2}}{\text {b1} \text {lambda1}}+e^{-\text {lambda2} y}\right )^{-\frac {\text {beta2}}{\text {lambda2}}}+\text {c3} \left (\frac {\text {b3} \left (-e^{-\text {lambda1} x}+e^{-\text {lambda1} K[1]}\right ) \text {lambda3}}{\text {b1} \text {lambda1}}+e^{-\text {lambda3} z}\right )^{-\frac {\text {beta3}}{\text {lambda3}}}+\text {c1} e^{\text {beta1} K[1]}\right )}{\text {b1}}dK[1]+c_1\left (\frac {\text {b2} e^{-\text {lambda1} x}}{\text {b1} \text {lambda1}}-\frac {e^{-\text {lambda2} y}}{\text {lambda2}},\frac {\text {b3} e^{-\text {lambda1} x}}{\text {b1} \text {lambda1}}-\frac {e^{-\text {lambda3} z}}{\text {lambda3}}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := b__1*exp(lambda__1*x)*diff(w(x,y,z),x)+ b__2*exp(lambda__2*y)*diff(w(x,y,z),y)+ b__3*exp(lambda__3*z)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*exp(beta__1*x)+ c__2*exp(beta__2*y)+ c__3*exp(beta__3*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {\left (f_{1} \left (\frac {-{\mathrm e}^{-\lambda _{2} y} b_{1} \lambda _{1} +{\mathrm e}^{-\lambda _{1} x} \lambda _{2} b_{2}}{b_{2} \lambda _{1} \lambda _{2}}, \frac {{\mathrm e}^{-\lambda _{1} x} \lambda _{3} b_{3} -{\mathrm e}^{-\lambda _{3} z} b_{1} \lambda _{1}}{b_{3} \lambda _{1} \lambda _{3}}\right ) b_{1} +\int _{}^{x}\left (c_{3} \left (\frac {b_{1} \lambda _{1}}{-{\mathrm e}^{-\lambda _{1} x} \lambda _{3} b_{3} +{\mathrm e}^{-\lambda _{3} z} b_{1} \lambda _{1} +{\mathrm e}^{-\lambda _{1} \textit {\_a}} b_{3} \lambda _{3}}\right )^{\frac {\beta _{3}}{\lambda _{3}}}+c_{1} {\mathrm e}^{\beta _{1} \textit {\_a}}+c_{2} \left (\frac {b_{1} \lambda _{1}}{{\mathrm e}^{-\lambda _{2} y} b_{1} \lambda _{1} -{\mathrm e}^{-\lambda _{1} x} \lambda _{2} b_{2} +{\mathrm e}^{-\lambda _{1} \textit {\_a}} b_{2} \lambda _{2}}\right )^{\frac {\beta _{2}}{\lambda _{2}}}\right ) {\mathrm e}^{\frac {-\lambda _{1}^{2} \textit {\_a} b_{1} +a \,{\mathrm e}^{-\lambda _{1} \textit {\_a}}}{b_{1} \lambda _{1}}}d \textit {\_a} \right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{-\lambda _{1} x}}{\lambda _{1} b_{1}}}}{b_{1}}\]

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6.9.5.7 [1960] Problem 7

problem number 1960

Added Jan 18, 2020.

Problem Chapter 9.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a_1 e^{\sigma _1 x+\beta _1 y} w_x + a_2 e^{\sigma _2 y+\beta _2 y} w_y + \left ( b_1 e^{\nu _1 x+\mu _1 y} + b_2 e^{\nu _2 x+\mu _2 y+ \lambda z} \right ) w_z = c_1 w +c_2 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Exp[sigma1*x+beta1*y]*D[w[x,y,z],x]+ a2*Exp[sigma2*y+beta2*y]*D[w[x,y,z],y]+( b1*Exp[nu1*x+mu1*y] +  b2*Exp[nu2*x+mu2*y+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(sigma__1*x+beta__1*y)*diff(w(x,y,z),x)+ a__2*exp(sigma__2*y+beta__2*y)*diff(w(x,y,z),y)+ ( b__1*exp(nu__1*x+mu__1*y) +  b__2*exp(nu__2*x+mu__2*y+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'));
 

time expired

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