6.6.16 6.3

6.6.16.1 [1514] Problem 1

problem number 1514

Added May 26, 2019.

Problem Chapter 6.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \tan (\gamma z) w_z = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, \frac {-x c \gamma +\ln \left (\operatorname {csgn}\left (\sec \left (\gamma z \right )\right ) \sin \left (\gamma z \right )\right ) a}{c \gamma }\right )\]

Hand solution

Solve

\begin{equation} aw_{x}+bw_{y}+c\tan \left ( \gamma z\right ) w_{z}=0 \tag {1}\end{equation}

Using Lagrange-charpit

\[ \frac {dx}{a}=\frac {dy}{b}=\frac {dz}{c\tan \left ( \gamma z\right ) }=\frac {dw}{0}\]

From first two pair of equation we obtain \(\frac {b}{a}x-y=C_{1}\) and from \(\frac {dx}{a}=\frac {dz}{c\tan \left ( \gamma z\right ) }\) we obtain \(\frac {c}{a}x-\frac {1}{\gamma }\ln \left ( \sin \left ( \gamma z\right ) \right ) =C_{2}\). Since \(dw=0\) and \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence

\[ u\left ( x,y,z\right ) =F\left ( \frac {b}{a}x-y,\frac {c}{a}x-\frac {1}{\gamma }\ln \left ( \sin \left ( \gamma z\right ) \right ) \right ) \]

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6.6.16.2 [1515] Problem 2

problem number 1515

Added May 26, 2019.

Problem Chapter 6.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-x b \beta +\ln \left (\operatorname {csgn}\left (\sec \left (\beta y \right )\right ) \sin \left (\beta y \right )\right ) a}{b \beta }, \frac {z a \lambda -\frac {c \ln \left (\sec \left (\lambda x \right )^{2}\right )}{2}}{a \lambda }\right )\]

Hand solution

Solve

\begin{equation} aw_{x}+b\tan \left ( \beta y\right ) w_{y}+c\tan \left ( \lambda x\right ) w_{z}=0 \tag {1}\end{equation}

Using Lagrange-charpit

\[ \frac {dx}{a}=\frac {dy}{b\tan \left ( \beta y\right ) }=\frac {dz}{c\tan \left ( \lambda x\right ) }=\frac {dw}{0}\]

From first two pair of equations, integrating gives \(\frac {b}{a}x-\frac {1}{\beta }\ln \left ( \sin \left ( \beta y\right ) \right ) =C_{1}\) and from \(\frac {dx}{a}=\frac {dz}{c\tan \left ( \lambda x\right ) }\) we obtain \(\frac {c}{a}\tan \left ( \lambda x\right ) dx=dz\). Integrating gives \(-\frac {c}{a\lambda }\ln \left ( \cos \left ( \lambda x\right ) \right ) -z=C_{2}\). Since \(dw=0\) then \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence

\[ u\left ( x,y,z\right ) =F\left ( \frac {b}{a}x-\frac {1}{\beta }\ln \left ( \sin \left ( \beta y\right ) \right ) ,-\frac {c}{a\lambda }\ln \left ( \cos \left ( \lambda x\right ) \right ) -z\right ) \]

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6.6.16.3 [1516] Problem 3

problem number 1516

Added May 26, 2019.

Problem Chapter 6.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.16.4 [1517] Problem 4

problem number 1517

Added May 26, 2019.

Problem Chapter 6.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.16.5 [1518] Problem 5

problem number 1518

Added May 26, 2019.

Problem Chapter 6.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \mu \nu \tan (\lambda x) w_x + \lambda \nu \tan (\mu y) w_y + \lambda \mu \tan (\nu z) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  mu*nu*Tan[lambda*x]*D[w[x, y,z], x] + lambda*nu*Tan[mu*y]*D[w[x, y,z], y] +lambda*mu*Tan[nu*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 :=  mu*nu*tan(lambda*x)*diff(w(x,y,z),x)+ lambda*nu*tan(mu*y)*diff(w(x,y,z),y)+lambda*mu*tan(nu*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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\ln \left (\operatorname {csgn}\left (\sec \left (\mu y \right )\right ) \csc \left (\lambda x \right ) \sin \left (\mu y \right )\right )}{\lambda }, \frac {\ln \left (\operatorname {csgn}\left (\sec \left (\nu z \right )\right ) \sin \left (\nu z \right ) \csc \left (\lambda x \right )\right )}{\lambda }\right )\]

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