6.6.17 6.4

6.6.17.1 [1519] Problem 1

problem number 1519

Added May 26, 2019.

Problem Chapter 6.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-y a +b x}{b}, \frac {-2 y c \gamma +b \ln \left (\csc \left (\gamma z \right )^{2}\right )-2 b \ln \left (\cot \left (\gamma z \right )\right )}{2 c \gamma }\right )\]

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6.6.17.2 [1520] Problem 2

problem number 1520

Added May 26, 2019.

Problem Chapter 6.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.17.3 [1521] Problem 3

problem number 1521

Added May 26, 2019.

Problem Chapter 6.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.17.4 [1522] Problem 4

problem number 1522

Added May 26, 2019.

Problem Chapter 6.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.6.17.5 [1523] Problem 5

problem number 1523

Added May 26, 2019.

Problem Chapter 6.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  mu*nu*Cot[lambda*x]*D[w[x, y,z], x] + lambda*nu*Cot[mu*y]*D[w[x, y,z], y] +lambda*mu*Cot[nu*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\begin{align*}& \left \{w(x,y,z)\to c_1\left (-\frac {\text {arccosh}\left (-\frac {1}{4} (\cos (2 \nu z)+3) \sec (\nu z)\right )}{\mu }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {\text {arccosh}\left (-\frac {1}{4} (\cos (2 \nu z)+3) \sec (\nu z)\right )}{\mu }\right )\right \}\\\end{align*}

Maple ✓

restart; 
pde :=  mu*nu*cot(lambda*x)*diff(w(x,y,z),x)+ lambda*nu*cot(mu*y)*diff(w(x,y,z),y)+lambda*mu*cot(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'));
 

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

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