Added June 26, 2019.
Problem Chapter 7.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \cot ^k(\lambda x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + c*D[w[x,y,z],z]== c*Cot[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)-\frac {c \cot ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)= c*cot(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \cot \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \cot (\gamma z) w_z = k \cot (\lambda x)+ s \cot (\beta y) \]
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]== k*Cot[lambda*x]+s*Cot[beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {a \lambda s \log (\tan (\beta y))+a \lambda s \log (\cos (\beta y))+b \beta k \log (\sin (\lambda x))}{a b \beta \lambda }+c_1\left (y-\frac {b x}{a},\frac {\log (\sec (\gamma z))}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; 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)= k*cot(lambda*x)+s*cot(beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{2\,\beta \,ba\lambda } \left ( 2\,{\it \_F1} \left ( {\frac {-ay+xb}{b}},1/2\,{\frac {-2\,c\gamma \,y+\ln \left ( \left ( \cot \left ( \gamma \,z \right ) \right ) ^{2}+1 \right ) b-2\,\ln \left ( \cot \left ( \gamma \,z \right ) \right ) b}{c\gamma }} \right ) \beta \,ba\lambda -k\ln \left ( \left ( \cot \left ( x\lambda \right ) \right ) ^{2}+1 \right ) \beta \,b-s\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) a\lambda \right ) }\]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cot ^n(\beta x) w_y + b \cot ^k(\lambda x) w_z = c \cot ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cot[beta*x]^n*D[w[x, y,z], y] + b*Cot[lambda*x]^k*D[w[x,y,z],z]== c*Cot[gamma*x]^m+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \cot ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }+z,\frac {a \cot ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\beta x)\right )}{\beta n+\beta }+y\right )-\frac {c \cot ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*cot(beta*x)^n*diff(w(x,y,z),y)+ b*cot(lambda*x)^k*diff(w(x,y,z),z)= c*cot(gamma*x)^m+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \cot \left ( x\gamma \right ) \right ) ^{m}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( \cot \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cot \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cot ^n(\lambda x) w_y + b \cot ^m(\beta y) w_z = c \cot ^k(\gamma y)+s \cot ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cot[lambda*x]^n*D[w[x, y,z], y] + b*Cot[beta*x]^m*D[w[x,y,z],z]== c*Cot[gamma*y]^k+s*Cot[mu*z]^r; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \cot ^k\left (\frac {\gamma \left (a \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda x)\right ) \cot ^{n+1}(\lambda x)+\lambda (n+1) y-a \cot ^{n+1}(\lambda K[1]) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda K[1])\right )\right )}{\lambda (n+1)}\right )+s \cot ^r\left (\frac {\mu \left (b \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\beta x)\right ) \cot ^{m+1}(\beta x)+\beta (m+1) z-b \cot ^{m+1}(\beta K[1]) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\beta K[1])\right )\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (\frac {b \cot ^{m+1}(\beta x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\beta x)\right )}{\beta m+\beta }+z,\frac {a \cot ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda x)\right )}{\lambda n+\lambda }+y\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*cot(lambda*x)^n*diff(w(x,y,z),y)+ b*cot(beta*x)^m*diff(w(x,y,z),z)= c*cot(gamma*y)^k+s*cot(mu*z)^r; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!c \left ( {\frac {-1+\cot \left ( \gamma \, \left ( -\int \!a \left ( \cot \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \cot \left ( \gamma \,a\int \! \left ( \cot \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) }{\cot \left ( \gamma \, \left ( -\int \!a \left ( \cot \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) +\cot \left ( \gamma \,a\int \! \left ( \cot \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) }} \right ) ^{k}+s \left ( {\frac {-1+\cot \left ( \mu \, \left ( -\int \!b \left ( \cot \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \right ) \cot \left ( \mu \,b\int \! \left ( \cot \left ( \beta \,{\it \_f} \right ) \right ) ^{m}\,{\rm d}{\it \_f} \right ) }{\cot \left ( \mu \, \left ( -\int \!b \left ( \cot \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \right ) +\cot \left ( \mu \,b\int \! \left ( \cot \left ( \beta \,{\it \_f} \right ) \right ) ^{m}\,{\rm d}{\it \_f} \right ) }} \right ) ^{r}{d{\it \_f}}+{\it \_F1} \left ( -\int \!a \left ( \cot \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cot \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cot (\beta x) w_y + c \cot (\lambda x) w_z = k \cot (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cot[beta*x]*D[w[x, y,z], y] + c*Cot[lambda*x]*D[w[x,y,z],z]== k*Cot[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \cot \left (\frac {\gamma (a \lambda z-c \log (\sin (\lambda x))+c \log (\sin (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b \log (\sin (\beta x))}{a \beta },z-\frac {c \log (\sin (\lambda x))}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*cot(beta*x)*diff(w(x,y,z),y)+ c*cot(lambda*x)*diff(w(x,y,z),z)= k*cot(gamma*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {k}{a} \left ( \cot \left ( {\frac {c\gamma \,\ln \left ( \left ( \cot \left ( {\it \_a}\,\lambda \right ) \right ) ^{2}+1 \right ) }{2\,a\lambda }} \right ) \cot \left ( {\frac {\gamma \, \left ( 2\,za\lambda +c\ln \left ( \left ( \cot \left ( x\lambda \right ) \right ) ^{2}+1 \right ) \right ) }{2\,a\lambda }} \right ) +1 \right ) \left ( \cot \left ( {\frac {c\gamma \,\ln \left ( \left ( \cot \left ( {\it \_a}\,\lambda \right ) \right ) ^{2}+1 \right ) }{2\,a\lambda }} \right ) -\cot \left ( {\frac {\gamma \, \left ( 2\,za\lambda +c\ln \left ( \left ( \cot \left ( x\lambda \right ) \right ) ^{2}+1 \right ) \right ) }{2\,a\lambda }} \right ) \right ) ^{-1}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {2\,ya\beta +b\ln \left ( \left ( \cot \left ( \beta \,x \right ) \right ) ^{2}+1 \right ) }{2\,a\beta }},{\frac {2\,za\lambda +c\ln \left ( \left ( \cot \left ( x\lambda \right ) \right ) ^{2}+1 \right ) }{2\,a\lambda }} \right ) \]
____________________________________________________________________________________