6.6.12 5.1

6.6.12.1 [1494] Problem 1

problem number 1494

Added May 26, 2019.

Problem Chapter 6.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \ln (\beta y) \ln (\lambda z) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.12.2 [1495] Problem 2

problem number 1495

Added May 26, 2019.

Problem Chapter 6.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \ln (\beta x) w_y + c \ln (\lambda x) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.12.3 [1496] Problem 3

problem number 1496

Added May 26, 2019.

Problem Chapter 6.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \ln (\beta x) \ln (\lambda y) w_y + c \ln (\mu x) \ln (\gamma z) w_z = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \int _1^y\frac {1}{\log (\lambda K[1])}dK[1]-b x \log (\beta x)+b x}{a},\frac {a \int _1^z\frac {1}{\log (\gamma K[2])}dK[2]-c x \log (\mu x)+c x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*ln(beta*x)*ln(lambda*y)*diff(w(x,y,z),y)+c*ln(mu*x)*ln(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'));
 

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

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6.6.12.4 [1497] Problem 4

problem number 1497

Added May 26, 2019.

Problem Chapter 6.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \ln (\beta x) w_x + b \ln (\lambda y) w_y + c \ln (\gamma z) w_z = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\int _1^y\frac {1}{\log (\lambda K[1])}dK[1]-\int _1^x\frac {b}{a \log (\beta K[2])}dK[2],\int _1^z\frac {1}{\log (\gamma K[3])}dK[3]-\int _1^x\frac {c}{a \log (\beta K[4])}dK[4]\right )\right \}\right \}\]

Maple ✓

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

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

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