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'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a y +b x}{b}, \frac {b \ln \left (\cot ^{2}\left (\gamma z \right )+1\right )-2 b \ln \left (\cot \left (\gamma z \right )\right )-2 \gamma c y}{2 \gamma c}\right )\]

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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 ) = \mathit {\_F1} \left (\frac {2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )}{2 b \beta }, \frac {-2 b \beta c \ln \left (\left (-\left (\cot ^{2}\left (\beta y \right )+1\right )^{\frac {i a \lambda }{b \beta }} \left ({\mathrm e}^{2 i \beta y}+1\right )^{\frac {2 i a \lambda }{b \beta }} \left (\cot ^{-\frac {2 i a \lambda }{b \beta }}\left (\beta y \right )\right ) {\mathrm e}^{-2 i \lambda x} {\mathrm e}^{\frac {\pi a \lambda \,\mathrm {csgn}\left (\frac {{\mathrm e}^{2 i \beta y}+1}{{\mathrm e}^{2 i \beta y}-1}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i \beta y}+1\right )}{{\mathrm e}^{2 i \beta y}-1}\right )}{b \beta }} {\mathrm e}^{\frac {\pi a \lambda \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \beta y}-1}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 i \beta y}+1\right )}{{\mathrm e}^{2 i \beta y}-1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 i \beta y}+i\right )}{b \beta }} {\mathrm e}^{-\frac {\pi a \lambda \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \beta y}-1}\right )}{b \beta }} {\mathrm e}^{-\frac {\pi a \lambda \,\mathrm {csgn}\left (\frac {{\mathrm e}^{2 i \beta y}+1}{{\mathrm e}^{2 i \beta y}-1}\right )}{b \beta }} {\mathrm e}^{-\frac {\pi a \lambda \,\mathrm {csgn}\left (i {\mathrm e}^{2 i \beta y}+i\right )}{b \beta }}+2^{\frac {2 i a \lambda }{b \beta }} \left ({\mathrm e}^{i \beta y}\right )^{\frac {2 i a \lambda }{b \beta }} {\mathrm e}^{-\frac {\pi \left (-\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{2 i \beta y}-1\right )^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 i \beta y}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 i \beta y}}{\left ({\mathrm e}^{2 i \beta y}-1\right )^{2}}\right )+\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{2 i \beta y}-1\right )^{2}}\right )-2 \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i \beta y}-1\right )\right )-2 \,\mathrm {csgn}\left (i \left (2 \,{\mathrm e}^{2 i \beta y}-{\mathrm e}^{4 i \beta y}-1\right )\right )+2 \,\mathrm {csgn}\left (i {\mathrm e}^{i \beta y}\right )-\mathrm {csgn}\left (i {\mathrm e}^{2 i \beta y}\right )+\mathrm {csgn}\left (\frac {i {\mathrm e}^{2 i \beta y}}{\left ({\mathrm e}^{2 i \beta y}-1\right )^{2}}\right )\right ) a \lambda }{2 b \beta }}\right ) \left ({\mathrm e}^{2 i \beta y}-1\right )^{-\frac {2 i a \lambda }{b \beta }}\right )+4 \left (-\frac {\pi c \,\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{2 i \beta y}-1\right )^{2}}\right )}{4}+\frac {\pi c \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i \beta y}-1\right )\right )}{2}-\frac {\pi c \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i \beta y}-1\right )^{2}\right )}{2}-\frac {\pi c \,\mathrm {csgn}\left (i {\mathrm e}^{i \beta y}\right )}{2}+\frac {\pi c \,\mathrm {csgn}\left (i {\mathrm e}^{2 i \beta y}\right )}{4}+\frac {\pi \left (\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{2 i \beta y}-1\right )^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 i \beta y}\right )-1\right ) c \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{2 i \beta y}}{\left ({\mathrm e}^{2 i \beta y}-1\right )^{2}}\right )}{4}-i c \ln \left ({\mathrm e}^{2 i \beta y}-1\right )+\frac {i c \ln \left ({\mathrm e}^{2 i \beta y}+1\right )}{2}+i c \ln \left ({\mathrm e}^{i \beta y}\right )+\frac {\left (b z +y c \right ) \beta }{2}+i \ln \left (2\right ) c \right ) a \lambda }{2 a b \beta \lambda }\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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b x}{a}+\frac {\log (\cos (\beta y))}{\beta },\frac {2 c \log (\cos (\beta y))}{\beta }-\frac {b \log \left (\cos ^2(\gamma z)\right )}{\gamma }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )}{2 b \beta }, \frac {-2 b \beta \ln \left (\frac {\cos \left (\gamma z \right )}{\sin \left (\gamma z \right )}\right )+b \beta \ln \left (\frac {\left (\frac {\cos \left (2 \beta y \right )}{2}+\frac {1}{2}\right )^{\frac {\gamma c}{b \beta }} \left (\cos ^{-\frac {2 \gamma c}{b \beta }}\left (\beta y \right )\right ) \left (\cos ^{2}\left (\gamma z \right )\right )}{\sin \left (\gamma z \right )^{2}}+1\right )+2 \gamma c \ln \left (\cos \left (\beta y \right )\right )}{2 \gamma \beta c}\right )\]

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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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b x}{a}+\frac {\log (\cos (\beta y))}{\beta },\frac {2 c \log (\cos (\beta y))}{\beta }-\frac {b \log \left (\cos ^2(\gamma z)\right )}{\gamma }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )}{2 b \beta }, \frac {-2 b \beta \ln \left (\frac {\cos \left (\gamma z \right )}{\sin \left (\gamma z \right )}\right )+b \beta \ln \left (\frac {\left (\frac {\cos \left (2 \beta y \right )}{2}+\frac {1}{2}\right )^{\frac {\gamma c}{b \beta }} \left (\cos ^{-\frac {2 \gamma c}{b \beta }}\left (\beta y \right )\right ) \left (\cos ^{2}\left (\gamma z \right )\right )}{\sin \left (\gamma z \right )^{2}}+1\right )+2 \gamma c \ln \left (\cos \left (\beta y \right )\right )}{2 \gamma \beta c}\right )\]

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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]];
 

Failed

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'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {\ln \left (\sqrt {\tan ^{2}\left (\mu y \right )+1}\, \cos \left (\lambda x \right )\right )}{\lambda }, \frac {\ln \left (\sqrt {\tan ^{2}\left (\nu z \right )+1}\, \cos \left (\lambda x \right )\right )}{\lambda }\right )\]

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