6.6.6 3.2

6.6.6.1 [1446] Problem 1

problem number 1446

Added May 18, 2019.

Problem Chapter 6.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*y*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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\int _1^ye^{-\beta K[1]} K[1]dK[1]+\frac {b e^{-\alpha x}}{a \alpha },-\frac {\gamma \int _1^x\frac {c e^{-\alpha K[2]}}{a \text {InverseFunction}\left [\int _1^{\text {$\#$1}}e^{-\beta K[1]} K[1]dK[1]\& \right ]\left [\frac {e^{-\alpha x} b-e^{-\alpha K[2]} b+a \alpha \int _1^ye^{-\beta K[1]} K[1]dK[1]}{a \alpha }\right ]}dK[2]+e^{-\gamma z}}{\gamma }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*y*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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {{\mathrm e}^{-\alpha x} b \,\beta ^{2}-{\mathrm e}^{-\beta y} a \alpha \left (\beta y +1\right )}{b \alpha \,\beta ^{2}}, \frac {\operatorname {LambertW}\left (-\left (\beta y +1\right ) {\mathrm e}^{-\beta y -1}\right ) {\mathrm e}^{-\gamma z} b \beta +{\mathrm e}^{-\beta y} c \gamma \left (\beta y +1\right )}{\alpha b \,\beta ^{2} c \gamma \operatorname {LambertW}\left (-\left (\beta y +1\right ) {\mathrm e}^{-\beta y -1}\right )}\right )\]

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6.6.6.2 [1447] Problem 2

problem number 1447

Added May 18, 2019.

Problem Chapter 6.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*y*Exp[alpha*x]*D[w[x, y,z], x] + b*Exp[beta*y]*D[w[x, y,z], y] +c*z*Exp[gamma*z]*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 :=  a*y*exp(alpha*x)*diff(w(x,y,z),x)+b*exp(beta*y)*diff(w(x,y,z),y)+c*z*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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-{\mathrm e}^{-\alpha x} b \,\beta ^{2}+{\mathrm e}^{-\beta y} a \alpha \left (\beta y +1\right )}{a \,\beta ^{2} \alpha }, \frac {{\mathrm e}^{-\beta y} c -\operatorname {Ei}_{1}\left (\gamma z \right ) b \beta }{c \beta }\right )\]

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6.6.6.3 [1448] Problem 3

problem number 1448

Added May 18, 2019.

Problem Chapter 6.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left [ y^2+ a e^{\alpha x}(\alpha -a e^{\alpha x}) \right ] w_y +\left [ z^2 +b z +c e^{\beta x}(\beta - b -c e^{\beta x}) \right ] w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + ( y^2+ a*Exp[alpha*x]*(alpha-a*Exp[alpha*x]))*D[w[x, y,z], y] +(z^2 +b*z +c*Exp[beta*x]*(beta - b -c*Exp[beta*x]))*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {2^{b/\beta } \beta ^{-\frac {b}{\beta }} e^{b x} c^{b/\beta } \left (L_{-\frac {b}{\beta }}^{\frac {b}{\beta }}\left (\frac {2 c e^{\beta x}}{\beta }\right ) \left (b-c e^{\beta x}+z\right )-2 c e^{\beta x} L_{-\frac {b+\beta }{\beta }}^{\frac {b+\beta }{\beta }}\left (\frac {2 c e^{\beta x}}{\beta }\right )\right )}{c e^{\beta x}-z},\frac {\left (y-a e^{\alpha x}\right ) \int _1^{e^{\alpha x}}\frac {e^{\frac {2 a K[1]}{\alpha }}}{K[1]}dK[1]+\alpha e^{\frac {2 a e^{\alpha x}}{\alpha }}}{a e^{\alpha x}-y}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+( y^2+ a*exp(alpha*x)*(alpha-a*exp(alpha*x)))*diff(w(x,y,z),y)+(z^2 +b*z +c*exp(beta*x)*(beta - b -c*exp(beta*x)))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \,{\mathrm e}^{\alpha x}+y}{\left (a \,{\mathrm e}^{\alpha x}-y \right ) \operatorname {Ei}_{1}\left (-\frac {2 a \,{\mathrm e}^{\alpha x}}{\alpha }\right )+{\mathrm e}^{\frac {2 a \,{\mathrm e}^{\alpha x}}{\alpha }} \alpha }, \frac {\left (-c \,{\mathrm e}^{\beta x}+z \right ) \int {\mathrm e}^{\frac {b x \beta +2 c \,{\mathrm e}^{\beta x}}{\beta }}d x +{\mathrm e}^{\frac {b x \beta +2 c \,{\mathrm e}^{\beta x}}{\beta }}}{c \,{\mathrm e}^{\beta x}-z}\right )\]

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6.6.6.4 [1449] Problem 4

problem number 1449

Added May 18, 2019.

Problem Chapter 6.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left [ y^2+ b y + a e^{\alpha x}(\alpha -b - a e^{\alpha x}) \right ] w_y +\left [ z^2 +c e^{\beta x}(z-k)-k^2 \right ] w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + ( y^2+ b*y + a*Exp[alpha*x]*(alpha-b-a*Exp[alpha*x]))*D[w[x, y,z], y] +(z^2 +c*Exp[beta*x]*(z-k)-k^2)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (2 k (-1)^{-\frac {k}{\beta }} \left (-\frac {\Gamma \left (\frac {2 k}{\beta },0,-\frac {c e^{\beta x}}{\beta }\right )}{\beta }+\frac {\beta ^{-\frac {2 k}{\beta }} c^{\frac {2 k}{\beta }} e^{\frac {c e^{\beta x}+2 \beta k x+2 i \pi k}{\beta }}}{k-z}\right ),\frac {2^{\frac {b}{\alpha }} \alpha ^{-\frac {b}{\alpha }} e^{b x} a^{\frac {b}{\alpha }} \left (L_{-\frac {b}{\alpha }}^{\frac {b}{\alpha }}\left (\frac {2 a e^{\alpha x}}{\alpha }\right ) \left (a \left (-e^{\alpha x}\right )+b+y\right )-2 a e^{\alpha x} L_{-\frac {\alpha +b}{\alpha }}^{\frac {\alpha +b}{\alpha }}\left (\frac {2 a e^{\alpha x}}{\alpha }\right )\right )}{a e^{\alpha x}-y}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+( y^2+ b*y+a*exp(alpha*x)*(alpha-b-a*exp(alpha*x)))*diff(w(x,y,z),y)+(z^2 +c*exp(beta*x)*(z-k)-k^2)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (-a \,{\mathrm e}^{\alpha x}+y \right ) \int {\mathrm e}^{\frac {b x \alpha +2 a \,{\mathrm e}^{\alpha x}}{\alpha }}d x +{\mathrm e}^{\frac {b x \alpha +2 a \,{\mathrm e}^{\alpha x}}{\alpha }}}{a \,{\mathrm e}^{\alpha x}-y}, \frac {\left (-z +k \right ) \int {\mathrm e}^{\frac {2 k x \beta +{\mathrm e}^{\beta x} c}{\beta }}d x -{\mathrm e}^{\frac {2 k x \beta +{\mathrm e}^{\beta x} c}{\beta }}}{-z +k}\right )\]

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6.6.6.5 [1450] Problem 5

problem number 1450

Added May 18, 2019.

Problem Chapter 6.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left [ y^2+ b y + a e^{\alpha x}(y-b)-b^2 \right ] w_y +\left [ z^2 +c(x z-1)e^{\beta x} \right ] w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + ( y^2+ b*y + a*Exp[alpha*x]*(y-b)-b^2)*D[w[x, y,z], y] +(z^2 +c*(x*z-1)*Exp[beta*x])*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)+( y^2+ b*y+a*exp(alpha*x)*(y-b)-b^2)*diff(w(x,y,z),y)+(z^2 +c*(x*z-1)*exp(beta*x))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.6.6.6 [1451] Problem 6

problem number 1451

Added May 18, 2019.

Problem Chapter 6.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left [ y^2 - a e^{\alpha x}(x y-1) \right ] w_y +\left (c e^{\beta x} z^2+ b e^{-\beta x} \right ) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + ( y^2- a*Exp[alpha*x]*(x*y-1))*D[w[x, y,z], y] +(c*Exp[beta*x]*z^2+b*Exp[-beta*x])*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)+( y^2- a*exp(alpha*x)*(x*y-1))*diff(w(x,y,z),y)+(c*exp(beta*x)*z^2+b*exp(-beta*x))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.6.6.7 [1452] Problem 7

problem number 1452

Added May 18, 2019.

Problem Chapter 6.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left ( a y^2 e^{\alpha x} + b e^{-\alpha x} \right ) w_y +\left [ d e^{\beta x} z^2+ c e^{\gamma x}(\gamma - c d e^{(\beta +\gamma )x} ) \right ] w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + ( a*y^2- a*Exp[alpha*x]+ b * Exp[-alpha*x])*D[w[x, y,z], y] +(d*Exp[beta*x]*z^2+c*Exp[gamma*x]*(gamma- c*d*Exp[(beta+gamma)*x]))*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)+( a*y^2- a*exp(alpha*x)+ b * exp(-alpha*x))*diff(w(x,y,z),y)+(d*exp(beta*x)*z^2+c*exp(gamma*x)*(gamma- c*d*exp((beta+gamma)*x)))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.6.6.8 [1453] Problem 8

problem number 1453

Added May 18, 2019.

Problem Chapter 6.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left [ b e^{\alpha x} y^2 + a e^{\beta x} (\beta - a b e^{(\alpha +\beta )x}) \right ] w_y +\left ( c z^2 e^{\gamma x}+ d z + k e^{-\gamma x} \right ) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (b*Exp[alpha*x]*y^2 + a*Exp[beta*x]*(beta- a*b*Exp[(alpha+beta)*x]))*D[w[x, y,z], y] +(c*z^2*Exp[gamma*x]+ d*z + k*Exp[-gamma*x])*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)+ (b*exp(alpha*x)*y^2 + a*exp(beta*x)*(beta- a*b*exp((alpha+beta)*x)))*diff(w(x,y,z),y)+(c*z^2*exp(gamma*x)+ d*z + k*exp(-gamma*x))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

time expired

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6.6.6.9 [1454] Problem 9

problem number 1454

Added May 18, 2019.

Problem Chapter 6.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left ( a e^{\alpha x} y^2 + b y + c e^{\alpha x} \right ) w_y +\left (e^{\beta x} z^2+ d e^{\gamma x}(z+\beta e^{-\beta x}) \right ) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*Exp[alpha*x]*y^2 + b*y +  c*Exp[alpha*x])*D[w[x, y,z], y] +(Exp[beta*x]*z^2+ d*Exp[gamma*x]*(z+beta*Exp[-beta*x]))*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)+ (a*exp(alpha*x)*y^2 + b*y +  c*exp(alpha*x))*diff(w(x,y,z),y)+(exp(beta*x)*z^2+ d*exp(gamma*x)*(z+beta*exp(-beta*x)))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\sqrt {a}\, y \operatorname {BesselJ}\left (-\frac {b +\alpha }{2 \alpha }, \frac {\sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\alpha x}}{\alpha }\right )-\sqrt {c}\, \operatorname {BesselJ}\left (\frac {-b +\alpha }{2 \alpha }, \frac {\sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\alpha x}}{\alpha }\right )}{-\sqrt {a}\, y \operatorname {BesselY}\left (-\frac {b +\alpha }{2 \alpha }, \frac {\sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\alpha x}}{\alpha }\right )+\sqrt {c}\, \operatorname {BesselY}\left (\frac {-b +\alpha }{2 \alpha }, \frac {\sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\alpha x}}{\alpha }\right )}, -\frac {{\mathrm e}^{\beta x} \left (\beta -\gamma \right ) \left ({\mathrm e}^{\beta x} z +\beta \right )}{\operatorname {hypergeom}\left (\left [\frac {-\beta +\gamma }{\gamma }\right ], \left [\frac {-\beta +2 \gamma }{\gamma }\right ], \frac {{\mathrm e}^{\gamma x} d}{\gamma }\right ) {\mathrm e}^{\gamma x} \beta d +z \,{\mathrm e}^{\beta x} \operatorname {hypergeom}\left (\left [-\frac {\beta }{\gamma }\right ], \left [\frac {-\beta +\gamma }{\gamma }\right ], \frac {{\mathrm e}^{\gamma x} d}{\gamma }\right ) \left (\beta -\gamma \right )}\right )\]

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6.6.6.10 [1455] Problem 10

problem number 1455

Added May 18, 2019.

Problem Chapter 6.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left [ e^{\alpha x} y^2 + a y e^{\beta x} + a \alpha e^{(\beta -\alpha )x} \right ] w_y +\left [ \gamma e^{\gamma x} z^2+ b e^{\delta x}(z+e^{-\gamma x}) \right ] w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + ( Exp[alpha*x]*y^2 + a*y*Exp[beta*x] + a*alpha*Exp[(beta-alpha)*x])*D[w[x, y,z], y] +(gamma*Exp[gamma*x]*z^2+ b*Exp[delta*x]*(z+Exp[-gamma*x]))*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)+ ( exp(alpha*x)*y^2 + a*y*exp(beta*x) + a*alpha*exp((beta-alpha)*x))*diff(w(x,y,z),y)+(gamma*exp(gamma*x)*z^2+ b*exp(delta*x)*(z+exp(-gamma*x)))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {{\mathrm e}^{\alpha x} \left (\beta -\alpha \right ) \left ({\mathrm e}^{\alpha x} y +\alpha \right )}{{\mathrm e}^{\beta x} \operatorname {hypergeom}\left (\left [\frac {\beta -\alpha }{\beta }\right ], \left [\frac {-\alpha +2 \beta }{\beta }\right ], \frac {a \,{\mathrm e}^{\beta x}}{\beta }\right ) a \alpha -y \,{\mathrm e}^{\alpha x} \operatorname {hypergeom}\left (\left [-\frac {\alpha }{\beta }\right ], \left [\frac {\beta -\alpha }{\beta }\right ], \frac {a \,{\mathrm e}^{\beta x}}{\beta }\right ) \left (\beta -\alpha \right )}, \frac {{\mathrm e}^{\gamma x} \left (\delta -\gamma \right ) \left (z \,{\mathrm e}^{\gamma x}+1\right )}{\operatorname {hypergeom}\left (\left [\frac {\delta -\gamma }{\delta }\right ], \left [\frac {2 \delta -\gamma }{\delta }\right ], \frac {{\mathrm e}^{\delta x} b}{\delta }\right ) {\mathrm e}^{\delta x} b -z \,{\mathrm e}^{\gamma x} \operatorname {hypergeom}\left (\left [-\frac {\gamma }{\delta }\right ], \left [\frac {\delta -\gamma }{\delta }\right ], \frac {{\mathrm e}^{\delta x} b}{\delta }\right ) \left (\delta -\gamma \right )}\right )\]

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6.6.6.11 [1456] Problem 11

problem number 1456

Added May 18, 2019.

Problem Chapter 6.3.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x+\left [ \alpha e^{\alpha x} y^2 + a e^{\beta x} (y+e^{-\alpha x}) \right ] w_y +\left [ e^{\gamma x} (z- b e^{\delta x})^2 + b \delta e^{\delta x} \right ] w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + ( alpha*Exp[alpha*x]*y^2 + a*Exp[beta*x]*(y+Exp[-alpha*x]))*D[w[x, y,z], y] +(Exp[gamma*x]*(z-b*Exp[delta*x])^2+b*delta*Exp[delta*x])*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)+ ( alpha*exp(alpha*x)*y^2 + a*exp(beta*x)*(y+exp(-alpha*x)))*diff(w(x,y,z),y)+(exp(gamma*x)*(z-b*exp(delta*x))^2+b*delta*exp(delta*x))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {{\mathrm e}^{\alpha x} \left (-\alpha +\beta \right ) \left (y \,{\mathrm e}^{\alpha x}+1\right )}{\operatorname {hypergeom}\left (\left [\frac {-\alpha +\beta }{\beta }\right ], \left [\frac {-\alpha +2 \beta }{\beta }\right ], \frac {{\mathrm e}^{\beta x} a}{\beta }\right ) a \,{\mathrm e}^{\beta x}-y \,{\mathrm e}^{\alpha x} \operatorname {hypergeom}\left (\left [-\frac {\alpha }{\beta }\right ], \left [\frac {-\alpha +\beta }{\beta }\right ], \frac {{\mathrm e}^{\beta x} a}{\beta }\right ) \left (-\alpha +\beta \right )}, \frac {b \,{\mathrm e}^{\delta x}-z}{\gamma \left ({\mathrm e}^{\gamma x} z -{\mathrm e}^{x \left (\delta +\gamma \right )} b +\gamma \right )}\right )\]

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6.6.6.12 [1457] Problem 12

problem number 1457

Added May 18, 2019.

Problem Chapter 6.3.2.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x+\left ( a_1 e^{\alpha x} y^2 + \beta y+ a_1 b_2^2 x^{2 \beta } e^{\alpha x} \right ) w_y +\left [ a_2 x^{2 n} z^2 e^{\lambda x}+(b_2 x^n e^{\lambda x} - n) z + c e^{\lambda x} \right ] w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y,z], x] + ( a1*Exp[alpha*x]*y^2 + beta*y+ a1*b2^2*x^(2*beta)*Exp[alpha*x])*D[w[x, y,z], y] +(a2*x^(2*n)*z^2*Exp[lamba*x]+(b2*x^n*Exp[lambda*x] - n)*z + c*Exp[lambda*x])*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 :=  x*diff(w(x,y,z),x)+ ( a1*exp(alpha*x)*y^2 + beta*y+ a1*b2^2*x^(2*beta)*exp(alpha*x))*diff(w(x,y,z),y)+(a2*x^(2*n)*z^2*exp(lamba*x)+(b2*x^n*exp(lambda*x) - n)*z + c*exp(lambda*x))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.6.6.13 [1458] Problem 13

problem number 1458

Added May 18, 2019.

Problem Chapter 6.3.2.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a_1 e^{\lambda _1 x} y + b_1 e^{\beta _1 x} y^k \right ) w_y + \left ( a_2 e^{\lambda _2 x} z + b_2 e^{\beta _1 x} z^m \right ) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = D[w[x, y,z], x] +( a1*Exp[lambda1*x]*y + b1*Exp[beta1*x]*y^k)*D[w[x, y,z], y] +(a2*Exp[lamba2*x]*z + b2*Exp[beta1*x]*z^m) *D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left ((k-1) \int _1^x\text {b1} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {beta1} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {a2} e^{\text {lamba2} K[2]} (m-1)}{\text {lamba2}}+\text {beta1} K[2]}dK[2]+z^{1-m} e^{\frac {\text {a2} (m-1) e^{\text {lamba2} x}}{\text {lamba2}}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ (a1*exp(lambda1*x)*y + b1*exp(beta1*x)*y^k)*diff(w(x,y,z),y)+(a2*exp(lamba2*x)*z + b2*exp(beta1*x)*z^m)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\operatorname {b1} \left (k -1\right ) \int {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\lambda \operatorname {1} x} \left (k -1\right )+\beta \operatorname {1} x \lambda \operatorname {1} }{\lambda \operatorname {1}}}d x +y^{1-k} {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\lambda \operatorname {1} x} \left (k -1\right )}{\lambda \operatorname {1}}}, \operatorname {b2} \left (m -1\right ) \int {\mathrm e}^{\frac {\operatorname {a2} \,{\mathrm e}^{\operatorname {lamba2} x} \left (m -1\right )+\beta \operatorname {1} x \operatorname {lamba2}}{\operatorname {lamba2}}}d x +z^{1-m} {\mathrm e}^{\frac {\operatorname {a2} \,{\mathrm e}^{\operatorname {lamba2} x} \left (m -1\right )}{\operatorname {lamba2}}}\right )\]

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6.6.6.14 [1459] Problem 14

problem number 1459

Added May 18, 2019.

Problem Chapter 6.3.2.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a_1 e^{\beta _1 x} y + b_1 e^{\gamma _1 x} y^k \right ) w_y + \left ( a_2 e^{\beta _2 x} + b_2 e^{\gamma _1 x+\lambda z} \right ) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = D[w[x, y,z], x] +( a1*Exp[beta1*x]*y + b1*Exp[gamma1*x]*y^k)*D[w[x, y,z], y] +(a2*Exp[beta2*x] + b2*Exp[gamma1*x+lambda*z]) *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(beta1*x)*y + b1*exp(gamma1*x)*y^k)*diff(w(x,y,z),y)+(a2*exp(beta2*x)+ b2*exp(gamma1*x+lambda*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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\operatorname {b1} \left (k -1\right ) \int {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\beta \operatorname {1} x} \left (k -1\right )+\gamma \operatorname {1} x \beta \operatorname {1} }{\beta \operatorname {1}}}d x +y^{1-k} {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\beta \operatorname {1} x} \left (k -1\right )}{\beta \operatorname {1}}}, \frac {-{\mathrm e}^{\frac {\lambda \left (\operatorname {a2} \,{\mathrm e}^{\beta \operatorname {2} x}-z \beta \operatorname {2} \right )}{\beta \operatorname {2}}}-\operatorname {b2} \int {\mathrm e}^{\frac {\lambda \operatorname {a2} \,{\mathrm e}^{\beta \operatorname {2} x}+\gamma \operatorname {1} x \beta \operatorname {2} }{\beta \operatorname {2}}}d x \lambda }{\lambda }\right )\]

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6.6.6.15 [1460] Problem 15

problem number 1460

Added May 18, 2019.

Problem Chapter 6.3.2.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a_1 x^n + b_1 x^m e^{\lambda y} \right ) w_y + \left ( a_2 x^k+b_2 x^s e^{\beta z} \right ) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = D[w[x, y,z], x] +( a1*x^n+b1*x^m*Exp[lambda*y])*D[w[x, y,z], y] +(a2*x^k+b2*x^2*Exp[beta*z]) *D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\text {b2} \beta x^3 \left (-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )^{-\frac {3}{k+1}} \Gamma \left (\frac {3}{k+1},-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\beta \left (-\text {a2} x^{k+1}+k z+z\right )}{k+1}}}{\text {a2} \text {b2} \beta ^2 \left (k^2-k-2\right )},\frac {(n+1) e^{-\frac {\lambda \left (-\text {a1} x^{n+1}+n y+y\right )}{n+1}}-\text {b1} \lambda x^{m+1} \left (-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \Gamma \left (\frac {m+1}{n+1},-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )}{\text {a1} \text {b1} \lambda ^2 (n+1) (m-n)}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ ( a1*x^n+b1*x^m*exp(lambda*y))*diff(w(x,y,z),y)+(a2*x^k+b2*x^2*exp(beta*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^{-n} \left (-\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )^{\frac {-m -n -2}{2 n +2}} \left (-{\mathrm e}^{\frac {x \lambda \operatorname {a1} \,x^{n}}{2 n +2}} \operatorname {b1} \,x^{m} \left (n +1\right )^{2} \left (-x \lambda \operatorname {a1} \,x^{n}+m +n +2\right ) \operatorname {WhittakerM}\left (\frac {m -n}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )+\left (-{\mathrm e}^{\frac {x \lambda \operatorname {a1} \,x^{n}}{2 n +2}} \operatorname {b1} \,x^{m} \left (n +1\right ) \left (m +n +2\right ) \operatorname {WhittakerM}\left (\frac {m +n +2}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )+{\mathrm e}^{-\frac {\left (-\operatorname {a1} x \,x^{n}+y \left (n +1\right )\right ) \lambda }{n +1}} \operatorname {a1} \,x^{n} \left (-\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )^{\frac {m +n +2}{2 n +2}} \left (m +1\right ) \left (m +2 n +3\right )\right ) \left (m +n +2\right )\right )}{\operatorname {a1} \lambda \left (m +1\right ) \left (m +2 n +3\right ) \left (m +n +2\right )}, \frac {\left (x^{2} \operatorname {b2} \,{\mathrm e}^{\frac {x^{k} x \beta \operatorname {a2}}{2 k +2}} \left (k +1\right )^{2} \left (-x^{k} x \beta \operatorname {a2} +k +4\right ) \operatorname {WhittakerM}\left (\frac {-k +2}{2 k +2}, \frac {2 k +5}{2 k +2}, -\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )+\left (x^{2} \operatorname {b2} \,{\mathrm e}^{\frac {x^{k} x \beta \operatorname {a2}}{2 k +2}} \left (k +4\right ) \left (k +1\right ) \operatorname {WhittakerM}\left (\frac {k +4}{2 k +2}, \frac {2 k +5}{2 k +2}, -\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )-6 \,{\mathrm e}^{-\frac {\left (-x^{k} \operatorname {a2} x +z \left (k +1\right )\right ) \beta }{k +1}} \left (k +\frac {5}{2}\right ) \operatorname {a2} \left (-\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )^{\frac {k +4}{2 k +2}} x^{k}\right ) \left (k +4\right )\right ) \left (-\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )^{\frac {-k -4}{2 k +2}} x^{-k}}{3 \operatorname {a2} \beta \left (2 k^{2}+13 k +20\right )}\right )\]

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6.6.6.16 [1461] Problem 16

problem number 1461

Added May 18, 2019.

Problem Chapter 6.3.2.16, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n e^{\lambda y} + b x y^m) w_x + e^{\mu y} w_y + \left ( c y^l z^k + d y^p z \right ) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*Exp[lambda*y] + b*x*y^m)*D[w[x, y,z], x] +Exp[mu*y]*D[w[x, y,z], y] +(c*y^L*z^k+d*y^p*z) *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 :=  (a*x^n*exp(lambda*y) + b*x*y^m)*diff(w(x,y,z),x)+ exp(mu*y)*diff(w(x,y,z),y)+(c*y^L*z^k+d*y^p*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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (a \left (n -1\right ) \int {\mathrm e}^{\frac {{\mathrm e}^{-\frac {\mu y}{2}} b \,y^{m} \left (\mu y \right )^{-\frac {m}{2}} \operatorname {WhittakerM}\left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) \left (n -1\right )-\mu y \left (-\lambda +\mu \right ) \left (m +1\right )}{\mu \left (m +1\right )}}d y +x^{1-n} {\mathrm e}^{\frac {{\mathrm e}^{-\frac {\mu y}{2}} b \,y^{m} \left (\mu y \right )^{-\frac {m}{2}} \operatorname {WhittakerM}\left (\frac {m}{2}, \frac {m}{2}+\frac {1}{2}, \mu y \right ) \left (n -1\right )}{\mu \left (m +1\right )}}, c \left (k -1\right ) \int y^{L} {\mathrm e}^{\frac {-\mu ^{2} \left (p +1\right ) y +d \,y^{p} \left (\mu y \right )^{-\frac {p}{2}} {\mathrm e}^{-\frac {\mu y}{2}} \operatorname {WhittakerM}\left (\frac {p}{2}, \frac {p}{2}+\frac {1}{2}, \mu y \right ) \left (k -1\right )}{\mu \left (p +1\right )}}d y +z^{1-k} {\mathrm e}^{\frac {d \,y^{p} \left (\mu y \right )^{-\frac {p}{2}} {\mathrm e}^{-\frac {\mu y}{2}} \operatorname {WhittakerM}\left (\frac {p}{2}, \frac {p}{2}+\frac {1}{2}, \mu y \right ) \left (k -1\right )}{\mu \left (p +1\right )}}\right )\]

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6.6.6.17 [1462] Problem 17

problem number 1462

Added May 18, 2019.

Problem Chapter 6.3.2.17, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (y^2 + 2 a \alpha e^{\alpha x^2}-a^2 e^{2 \alpha x^2} )w_y + \left ( c e^{-2 \beta x^2} z^2 + 2 \beta x z + b^2 c \right ) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (y^2 + 2*a*alpha*Exp[alpha*x^2]-a^2*Exp[2*alpha*x^2] )*D[w[x, y,z], y] +( c*Exp[-2*beta*x^2]*z^2 + 2*beta*x*z + b^2*c)*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)+ (y^2 + 2*a*alpha*exp(alpha*x^2)-a^2*exp(2*alpha*x^2) )*diff(w(x,y,z),y)+( c*exp(-2*beta*x^2)*z^2 + 2*beta*x*z + b^2*c)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.6.6.18 [1463] Problem 18

problem number 1463

Added May 18, 2019.

Problem Chapter 6.3.2.18, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a e^{2 \alpha x^2} y^2 + 2 \alpha x y + a b^2 )w_y + \left ( c x^\beta z^2 + 2 \gamma x z + c d^2 x^\beta e^{2\gamma x^2} \right ) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*Exp[2*alpha*x^2]*y^2 + 2*alpha*x*y + a*b^2 )*D[w[x, y,z], y] +( c*x^beta*z^2 + 2*gamma*x*z + c*d^2*x^beta*Exp[2*gamma*x^2])*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)+ (a*exp(2*alpha*x^2)*y^2 + 2*alpha*x*y + a*b^2 )*diff(w(x,y,z),y)+( c*x^beta*z^2 + 2*gamma*x*z + c*d^2*x^beta*exp(2*gamma*x^2))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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