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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-ay+xb}{b}},{\frac {-2\,yc\gamma -2\,b\ln \left ( \cot \left ( \gamma \,z \right ) \right ) +b\ln \left ( \left ( \cot \left ( \gamma \,z \right ) \right ) ^{2}+1 \right ) }{2\,c\gamma }} \right ) \]
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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 ( {\frac {2\,b\beta \,x+2\,\ln \left ( \cot \left ( \beta \,y \right ) \right ) a-a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2\,\beta \,b}},{\frac {1}{2\,\beta \,ba\lambda } \left ( -2\,c\ln \left ( \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{{\frac {-2\,ia\lambda }{\beta \,b}}} \left ( {{\rm e}^{-1/2\,{\frac {a\pi \,\lambda }{\beta \,b} \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 \left ( -1-{{\rm e}^{4\,i\beta \,y}}+2\,{{\rm e}^{2\,i\beta \,y}} \right ) \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{{\rm e}^{i\beta \,y}} \right ) \right ) }}}{2}^{{\frac {2\,ia\lambda }{\beta \,b}}} \left ( {{\rm e}^{i\beta \,y}} \right ) ^{{\frac {2\,ia\lambda }{\beta \,b}}}-{\frac {1}{ \left ( {{\rm e}^{i\lambda \,x}} \right ) ^{2}}{{\rm e}^{{\frac {\lambda \,{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}}+i \right ) \pi \,a}{\beta \,b}{\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 {a\pi \,\lambda }{\beta \,b}{\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 }{\beta \,b}}} \left ( {{\rm e}^{2\,i\beta \,y}}+1 \right ) ^{{\frac {2\,ia\lambda }{\beta \,b}}} \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{{\frac {ia\lambda }{\beta \,b}}} \right ) ^{-2} \left ( {{\rm e}^{{\frac {\lambda \,{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}}+i \right ) \pi \,a}{\beta \,b}}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {a\pi \,\lambda }{\beta \,b}{\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 {a\pi \,\lambda }{\beta \,b}{\it csgn} \left ( {\frac {i}{{{\rm e}^{2\,i\beta \,y}}-1}} \right ) }}} \right ) ^{-1}} \right ) \right ) \beta \,b+4\,\lambda \, \left ( 1/4\,c\pi \, \left ( {\it csgn} \left ( {\frac {i}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) {\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}} \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\,c{\it csgn} \left ( {\frac {i}{ \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2}}} \right ) \pi -1/2\,c{\it csgn} \left ( i \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) ^{2} \right ) \pi +1/2\,c{\it csgn} \left ( i \left ( {{\rm e}^{2\,i\beta \,y}}-1 \right ) \right ) \pi +1/4\,c{\it csgn} \left ( i{{\rm e}^{2\,i\beta \,y}} \right ) \pi -1/2\,c{\it csgn} \left ( i{{\rm e}^{i\beta \,y}} \right ) \pi -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+yc \right ) \right ) a \right ) } \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {2\,b\beta \,x+2\,\ln \left ( \cot \left ( \beta \,y \right ) \right ) a-a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2\,\beta \,b}},{\frac {1}{2\,\beta \,\gamma \,c} \left ( \ln \left ( {\frac { \left ( \cot \left ( \gamma \,z \right ) \right ) ^{2}+1}{ \left ( \cot \left ( \gamma \,z \right ) \right ) ^{2}}} \right ) \beta \,b+c\gamma \,\ln \left ( \left ( \cos \left ( \beta \,y \right ) \right ) ^{2} \right ) \right ) } \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {2\,b\beta \,x+2\,\ln \left ( \cot \left ( \beta \,y \right ) \right ) a-a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2\,\beta \,b}},{\frac {1}{2\,\beta \,\gamma \,c} \left ( \ln \left ( {\frac { \left ( \cot \left ( \gamma \,z \right ) \right ) ^{2}+1}{ \left ( \cot \left ( \gamma \,z \right ) \right ) ^{2}}} \right ) \beta \,b+c\gamma \,\ln \left ( \left ( \cos \left ( \beta \,y \right ) \right ) ^{2} \right ) \right ) } \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\lambda }\ln \left ( \sqrt {1+ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}}\cos \left ( x\lambda \right ) \right ) },{\frac {1}{\lambda }\ln \left ( \sqrt {1+ \left ( \tan \left ( \nu \,z \right ) \right ) ^{2}}\cos \left ( x\lambda \right ) \right ) } \right ) \]
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