6.6.12 5.1

6.6.12.1 [1495] Problem 1

problem number 1495

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]];
 

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

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-ya+bx}{b}},{\frac {-b\Ei \left ( 1,-\ln \left ( \lambda \,z \right ) \right ) -y\lambda \,c \left ( \ln \left ( \beta \,y \right ) -1 \right ) }{\lambda \,c}} \right ) \]

____________________________________________________________________________________

6.6.12.2 [1496] Problem 2

problem number 1496

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-bx\ln \left ( \beta \,x \right ) +ya+bx}{a}},{\frac {-cx\ln \left ( \lambda \,x \right ) +za+cx}{a}} \right ) \]

____________________________________________________________________________________

6.6.12.3 [1497] Problem 3

problem number 1497

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\text {li}(\lambda y)}{\lambda }-\frac {b x (\log (\beta x)-1)}{a},\frac {\text {li}(\gamma z)}{\gamma }-\frac {c x (\log (\mu x)-1)}{a}\right )\right \}\right \}\]

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 ) ={\it \_F1} \left ( {\frac {\Ei \left ( 1,-\ln \left ( y\lambda \right ) \right ) a+x\lambda \,b \left ( \ln \left ( \beta \,x \right ) -1 \right ) }{a\lambda }},-8\,{\frac {b}{ca\beta \,\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) } \left ( -1/8\,\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) c \left ( \ln \left ( {\frac {\mu \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) }{\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) -\ln \left ( {\frac {\beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) }{\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) \right ) \left ( \LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) -\ln \left ( {\frac {\beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) }{\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) +1 \right ) \Ei \left ( 1,-\ln \left ( {\frac {\beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) }{\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) \right ) +\beta \, \left ( -1/8\,xc \left ( \ln \left ( \beta \,x \right ) -1 \right ) \ln \left ( {\frac {\beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) }{\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) +1/8\,xc \left ( \ln \left ( \beta \,x \right ) -1 \right ) \ln \left ( {\frac {\mu \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) }{\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) +\LambertW \left ( \beta \,x \left ( \ln \left ( \beta \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) \left ( a\Ei \left ( 1,3\,\ln \left ( 2 \right ) -\ln \left ( z \right ) \right ) +1/8\,xc \left ( \ln \left ( \beta \,x \right ) -1 \right ) \right ) \right ) \right ) } \right ) \]

____________________________________________________________________________________

6.6.12.4 [1498] Problem 4

problem number 1498

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]];
 

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

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-\Ei \left ( 1,-\ln \left ( y\lambda \right ) \right ) a\beta +\Ei \left ( 1,-\ln \left ( \beta \,x \right ) \right ) b\lambda }{b\beta \,\lambda }},{\frac {-8\,a\Ei \left ( 1,3\,\ln \left ( 2 \right ) -\ln \left ( z \right ) \right ) \beta +\Ei \left ( 1,-\ln \left ( \beta \,x \right ) \right ) c}{\beta \,c}} \right ) \]

____________________________________________________________________________________