6.6.17 6.4

6.6.17.1 [1520] Problem 1

problem number 1520

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {4\,\ln \left ( \left ( \cot \left ( z/8 \right ) \right ) ^{2}+1 \right ) a-8\,a\ln \left ( \cot \left ( z/8 \right ) \right ) -cx}{c}} \right ) \]

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6.6.17.2 [1521] Problem 2

problem number 1521

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 ) ={\it \_F1} \left ( 1/2\,{\frac {2\,bx\beta +2\,a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{b\beta }},1/2\,{\frac {1}{b\beta \,a\lambda } \left ( -2\,c\ln \left ( \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{{\frac {-2\,ia\lambda }{b\beta }}} \left ( {2}^{{\frac {2\,ia\lambda }{b\beta }}} \left ( {{\rm e}^{i\beta \,y}} \right ) ^{{\frac {2\,ia\lambda }{b\beta }}}{{\rm e}^{-1/2\,{\frac {\pi \,\lambda \,a}{b\beta } \left ( -{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}} \right ) {\it csgn} \left ( {\frac {i{{\rm e}^{2\,i\beta \,y}}}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) {\it csgn} \left ( {\frac {i}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) -2\,{\it csgn} \left ( i \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) \right ) +2\,{\it csgn} \left ( i{{\rm e}^{i\beta \,y}} \right ) -{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}} \right ) +{\it csgn} \left ( {\frac {i{{\rm e}^{2\,i\beta \,y}}}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) +{\it csgn} \left ( {\frac {i}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) +2\,{\it csgn} \left ( i \left ( {{\rm e}^{4\,i\beta \,y}}-2\,{{\rm e}^{2\,i\beta \,y}}+1 \right ) \right ) \right ) }}}-{\frac {1}{ \left ( {{\rm e}^{i\lambda \,x}} \right ) ^{2}} \left ( {{\rm e}^{2\,i\beta \,y}}+1 \right ) ^{{\frac {2\,ia\lambda }{b\beta }}}{{\rm e}^{{\frac {\pi \,\lambda \,{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}}+i \right ) a}{b\beta }{\it csgn} \left ( {\frac {i \left ( {{\rm e}^{2\,i\beta \,y}}+1 \right ) }{{{\rm e}^{2\,i\beta \,y}}-1}} \right ) {\it csgn} \left ( {\frac {i}{{{\rm e}^{2\,i\beta \,y}}-1}} \right ) }}}{{\rm e}^{{\frac {\pi \,\lambda \,a}{b\beta }{\it csgn} \left ( {\frac {i \left ( {{\rm e}^{2\,i\beta \,y}}+1 \right ) }{{{\rm e}^{2\,i\beta \,y}}-1}} \right ) {\it csgn} \left ( {\frac {{{\rm e}^{2\,i\beta \,y}}+1}{{{\rm e}^{2\,i\beta \,y}}-1}} \right ) }}} \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) ^{{\frac {ia\lambda }{b\beta }}} \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{{\frac {ia\lambda }{b\beta }}} \right ) ^{-2} \left ( {{\rm e}^{{\frac {\pi \,\lambda \,{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}}+i \right ) a}{b\beta }}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {\pi \,\lambda \,a}{b\beta }{\it csgn} \left ( {\frac {{{\rm e}^{2\,i\beta \,y}}+1}{{{\rm e}^{2\,i\beta \,y}}-1}} \right ) }}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {\pi \,\lambda \,a}{b\beta }{\it csgn} \left ( {\frac {i}{{{\rm e}^{2\,i\beta \,y}}-1}} \right ) }}} \right ) ^{-1}} \right ) \right ) b\beta +4\, \left ( 1/4\,c\pi \, \left ( {\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}} \right ) {\it csgn} \left ( {\frac {i}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) -1 \right ) {\it csgn} \left ( {\frac {i{{\rm e}^{2\,i\beta \,y}}}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) -1/4\,\pi \,c{\it csgn} \left ( {\frac {i}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) -1/2\,\pi \,c{\it csgn} \left ( i \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2} \right ) +1/2\,\pi \,c{\it csgn} \left ( i \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) \right ) +1/4\,\pi \,c{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}} \right ) -1/2\,\pi \,c{\it csgn} \left ( i{{\rm e}^{i\beta \,y}} \right ) -i\ln \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) c+i/2\ln \left ( {{\rm e}^{2\,i\beta \,y}}+1 \right ) c+i\ln \left ( {{\rm e}^{i\beta \,y}} \right ) c+i\ln \left ( 2 \right ) c+1/2\,\beta \, \left ( bz+cy \right ) \right ) \lambda \,a \right ) } \right ) \]

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6.6.17.3 [1522] Problem 3

problem number 1522

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( 1/2\,{\frac {2\,bx\beta +2\,a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{b\beta }},8\,{\frac {b}{c}\ln \left ( {\frac {1}{\cot \left ( z/8 \right ) }\sqrt { \left ( \left ( \cos \left ( \beta \,y \right ) \right ) ^{2} \right ) ^{1/8\,{\frac {c}{b\beta }}} \left ( \cot \left ( z/8 \right ) \right ) ^{2}+ \left ( \left ( \cos \left ( \beta \,y \right ) \right ) ^{2} \right ) ^{1/8\,{\frac {c}{b\beta }}}}} \right ) } \right ) \]

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6.6.17.4 [1523] Problem 4

problem number 1523

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( 1/2\,{\frac {2\,bx\beta +2\,a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{b\beta }},8\,{\frac {b}{c}\ln \left ( {\frac {1}{\cot \left ( z/8 \right ) }\sqrt { \left ( \left ( \cos \left ( \beta \,y \right ) \right ) ^{2} \right ) ^{1/8\,{\frac {c}{b\beta }}} \left ( \cot \left ( z/8 \right ) \right ) ^{2}+ \left ( \left ( \cos \left ( \beta \,y \right ) \right ) ^{2} \right ) ^{1/8\,{\frac {c}{b\beta }}}}} \right ) } \right ) \]

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6.6.17.5 [1524] Problem 5

problem number 1524

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'));
 

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

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