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]];
 

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

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a \alpha y -b \,{\mathrm e}^{\alpha x}}{a \alpha }, \frac {a \alpha z +c \Ei \left (1, -\frac {b \beta \,{\mathrm e}^{\alpha x}}{a \alpha }\right ) {\mathrm e}^{\frac {\left (a \alpha y -b \,{\mathrm e}^{\alpha x}\right ) \beta }{a \alpha }}}{a \alpha }\right )\]

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\alpha 1 x}}{\alpha 1}+\nu 1 x}d x \right )-{\mathrm e}^{\frac {\left (\mathit {A1} \,{\mathrm e}^{\alpha 1 x}-\alpha 1 y \right ) \lambda }{\alpha 1}}}{\lambda }, z -\left (\int _{}^{x}\left (\mathit {B2} \left (-\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\mathit {\_b} \alpha 1}+\mathit {\_b} \alpha 1 \nu 1}{\alpha 1}}d \mathit {\_b} \right )+\mathit {B1} \lambda \left (\int {\mathrm e}^{\frac {\mathit {A1} \lambda \,{\mathrm e}^{\alpha 1 x}}{\alpha 1}+\nu 1 x}d x \right )+{\mathrm e}^{\frac {\left (\mathit {A1} \,{\mathrm e}^{\alpha 1 x}-\alpha 1 y \right ) \lambda }{\alpha 1}}\right )^{-\frac {\beta }{\lambda }} {\mathrm e}^{\frac {\mathit {A1} \beta \,{\mathrm e}^{\mathit {\_b} \alpha 1}+\mathit {\_b} \alpha 1 \nu 2}{\alpha 1}}+\mathit {A2} \,{\mathrm e}^{\mathit {\_b} \alpha 2}\right )d\mathit {\_b} \right )\right )\]

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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'));
 

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

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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'));
 

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

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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]];
 

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

Maple ✓

restart; 
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'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-\mathit {a1} \alpha \RootOf \left (\mathit {a1} \alpha \beta y -\mathit {a1} \alpha \ln \left (\frac {\left (-\mathit {b1} +{\mathrm e}^{\mathit {\_Z}}\right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\mathit {b1} \beta }{\mathit {a1} \alpha }}}{\mathit {b2}}\right )+\mathit {b1} \beta \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )-\left (-\mathit {b1} \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )+\left (-y \mathit {a1} +\mathit {b1} x \right ) \alpha \right ) \beta }{\mathit {a1} \alpha \mathit {b1} \beta }, \frac {-\mathit {a1} \alpha \RootOf \left (\gamma \mathit {a1} \alpha z -\mathit {a1} \alpha \ln \left (\frac {\left (-\mathit {c1} +{\mathrm e}^{\mathit {\_Z}}\right ) \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )^{\frac {\gamma \mathit {c1}}{\mathit {a1} \alpha }}}{\mathit {c2}}\right )+\gamma \mathit {c1} \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )\right )-\left (-\mathit {c1} \ln \left (\mathit {a2} \,{\mathrm e}^{\alpha x}+\mathit {a1} \right )+\left (-\mathit {a1} z +\mathit {c1} x \right ) \alpha \right ) \gamma }{\gamma \mathit {a1} \alpha \mathit {c1}}\right )\]

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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]];
 

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

Maple ✓

restart; 
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'));
 

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

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