6.6.5 3.1

6.6.5.1 [1438] Problem 1

problem number 1438

Added May 18, 2019.

Problem Chapter 6.3.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x+ b e^{\alpha x} w_y +c e^{\beta y} w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] +b*Exp[alpha*x]*D[w[x, y,z], y] +c*Exp[beta*y]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+b*exp(alpha*x)*diff(w(x,y,z),y)+c*exp(beta*y)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.5.2 [1439] Problem 2

problem number 1439

Added May 18, 2019.

Problem Chapter 6.3.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x+ b e^{\alpha x} w_y +c e^{\gamma z} w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.5.3 [1440] Problem 3

problem number 1440

Added May 18, 2019.

Problem Chapter 6.3.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x+ b e^{\beta y} w_y +c e^{\gamma z} w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.5.4 [1441] Problem 4

problem number 1441

Added May 18, 2019.

Problem Chapter 6.3.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+(A_1 e^{\alpha _1 x} + B_1 e^{\nu _1 x+\lambda y}) w_y + (A_2 e^{\alpha _2 x} + B_2 e^{\nu _2 x+\beta y}) w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-{\mathrm e}^{\frac {\lambda \left (\operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} x}-\alpha \operatorname {1} y \right )}{\alpha \operatorname {1}}}-\operatorname {B1} \int {\mathrm e}^{\frac {\lambda \operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} x}+\nu \operatorname {1} x \alpha \operatorname {1} }{\alpha \operatorname {1}}}d x \lambda }{\lambda }, -\int _{}^{x}\left (\operatorname {A2} \,{\mathrm e}^{\alpha \operatorname {2} \textit {\_b}}+\operatorname {B2} \left (\operatorname {B1} \int {\mathrm e}^{\frac {\lambda \operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} x}+\nu \operatorname {1} x \alpha \operatorname {1} }{\alpha \operatorname {1}}}d x \lambda -\operatorname {B1} \int {\mathrm e}^{\frac {\lambda \operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} \textit {\_b}}+\textit {\_b} \alpha \operatorname {1} \nu \operatorname {1} }{\alpha \operatorname {1}}}d \textit {\_b} \lambda +{\mathrm e}^{\frac {\lambda \left (\operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} x}-\alpha \operatorname {1} y \right )}{\alpha \operatorname {1}}}\right )^{-\frac {\beta }{\lambda }} {\mathrm e}^{\frac {\beta \operatorname {A1} \,{\mathrm e}^{\alpha \operatorname {1} \textit {\_b}}+\nu \operatorname {2} \textit {\_b} \alpha \operatorname {1} }{\alpha \operatorname {1}}}\right )d \textit {\_b} +z \right )\]

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6.6.5.5 [1442] Problem 5

problem number 1442

Added May 18, 2019.

Problem Chapter 6.3.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\alpha x}w_x+ b e^{\beta y} w_y + c e^{\gamma z} w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.5.6 [1443] Problem 6

problem number 1443

Added May 18, 2019.

Problem Chapter 6.3.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\beta y}w_x+ b e^{\alpha x} w_y + c e^{\gamma z} w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[beta*y]*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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.6.5.7 [1444] Problem 7

problem number 1444

Added May 18, 2019.

Problem Chapter 6.3.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a_1+ a_2 e^{\alpha x}) w_x+ (b_1 + b_2 e^{\beta y} w_y + (c_1+c_2 e^{\gamma z}) w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-x \beta \operatorname {b1} +\ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\beta \operatorname {b1}}{\operatorname {a1} \alpha }} \left (-\operatorname {b1} +\operatorname {RootOf}\left (y \alpha \operatorname {a1} \beta -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\beta \operatorname {b1}}{\operatorname {a1} \alpha }} \left (-\operatorname {b1} +\textit {\_Z} \right )}{\operatorname {b2}}\right )+\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \beta \operatorname {b1} \right )\right )}{\operatorname {b2} \operatorname {RootOf}\left (y \alpha \operatorname {a1} \beta -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\beta \operatorname {b1}}{\operatorname {a1} \alpha }} \left (-\operatorname {b1} +\textit {\_Z} \right )}{\operatorname {b2}}\right )+\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \beta \operatorname {b1} \right )}\right ) \operatorname {a1}}{\operatorname {a1} \operatorname {b1} \beta }, \frac {-\operatorname {c1} \gamma x +\ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\gamma \operatorname {c1}}{\alpha \operatorname {a1}}} \left (-\operatorname {c1} +\operatorname {RootOf}\left (z \alpha \operatorname {a1} \gamma -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\gamma \operatorname {c1}}{\alpha \operatorname {a1}}} \left (-\operatorname {c1} +\textit {\_Z} \right )}{\operatorname {c2}}\right )+\gamma \operatorname {c1} \ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )\right )\right )}{\operatorname {c2} \operatorname {RootOf}\left (z \alpha \operatorname {a1} \gamma -\alpha \operatorname {a1} \ln \left (\frac {\left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )^{\frac {\gamma \operatorname {c1}}{\alpha \operatorname {a1}}} \left (-\operatorname {c1} +\textit {\_Z} \right )}{\operatorname {c2}}\right )+\gamma \operatorname {c1} \ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )\right )}\right ) \operatorname {a1}}{\operatorname {a1} \operatorname {c1} \gamma }\right )\]

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6.6.5.8 [1445] Problem 8

problem number 1445

Added May 18, 2019.

Problem Chapter 6.3.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ e^{\beta y} (a_1+ a_2 e^{\alpha x}) w_x+ e^{\alpha x}(b_1 + b_2 e^{\beta y} w_y +c e^{\beta y + \gamma z} w_z = 0 \]

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]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
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)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-x \beta \operatorname {b2} +\ln \left (\frac {\operatorname {RootOf}\left (y \alpha \operatorname {a2} \beta +\alpha x \beta \operatorname {b2} -\alpha \operatorname {a2} \ln \left (\frac {\operatorname {b1} \left (\operatorname {a2} +{\mathrm e}^{-\alpha x} \operatorname {a1} \right )^{-\frac {\beta \operatorname {b2}}{\alpha \operatorname {a2}}}}{-\operatorname {b2} +\textit {\_Z}}\right )-\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \operatorname {b2} \beta \right ) \operatorname {b1} \left (\operatorname {a2} +{\mathrm e}^{-\alpha x} \operatorname {a1} \right )^{-\frac {\beta \operatorname {b2}}{\alpha \operatorname {a2}}}}{-\operatorname {b2} +\operatorname {RootOf}\left (y \alpha \operatorname {a2} \beta +\alpha x \beta \operatorname {b2} -\alpha \operatorname {a2} \ln \left (\frac {\operatorname {b1} \left (\operatorname {a2} +{\mathrm e}^{-\alpha x} \operatorname {a1} \right )^{-\frac {\beta \operatorname {b2}}{\alpha \operatorname {a2}}}}{-\operatorname {b2} +\textit {\_Z}}\right )-\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right ) \operatorname {b2} \beta \right )}\right ) \operatorname {a2}}{\operatorname {a2} \operatorname {b2} \beta }, \frac {-{\mathrm e}^{-\gamma z} \alpha \operatorname {a1} +c \gamma \left (\ln \left (\operatorname {a1} +\operatorname {a2} \,{\mathrm e}^{\alpha x}\right )-\alpha x \right )}{\operatorname {a1} \alpha c \gamma }\right )\]

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