Added Feb. 11, 2019.
Problem Chapter 3.6.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b y^n w_y = c \cot (\lambda x) + k \cot (\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Cot[lambda*x] + k*Cot[mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {a k \lambda \log (\tan (\mu y))+a k \lambda \log (\cos (\mu y))+b c \mu \log (\sin (\lambda x))}{a b \lambda \mu }\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*cot(lambda*x)+k*cot(mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =1/2\,{\frac {1}{b\mu \,a\lambda } \left ( 2\,{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda -k\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) a\lambda -c\ln \left ( \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) b\mu \right ) }\]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.6.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b y^n w_y = c \cot (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Cot[lambda*x + mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c (\log (\tan (\lambda x+\mu y))+\log (\cos (\lambda x+\mu y)))}{a \lambda +b \mu }\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*cot(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =-{\frac {c\ln \left ( \left ( \cot \left ( \lambda \,x+\mu \,y \right ) \right ) ^{2}+1 \right ) }{2\,a\lambda +2\,b\mu }}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.6.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x \cot (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Cot[lambda*x + mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right )+\frac {a x (\log (\tan (\lambda x+\mu y))+\log (\cos (\lambda x+\mu y)))}{\lambda x+\mu y}\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x) + y*diff(w(x,y),y) = a*x*cot(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =-1/2\,{a\ln \left ( \left ( \cot \left ( \lambda \,x+\mu \,y \right ) \right ) ^{2}+1 \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.6.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \cot ^n(\lambda x) w_y = c\cot ^m(\mu x)+s \cot ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Cot[lambda*x]^n*D[w[x, y], y] == c*Cot[mu*x]^m + s*Cot[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \cot ^k\left (\frac {\beta \left (b \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda x)\right ) \cot ^{n+1}(\lambda x)+a \lambda (n+1) y-b \cot ^{n+1}(\lambda K[1]) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda K[1])\right )\right )}{a \lambda (n+1)}\right )+c \cot ^m(\mu K[1])}{a}dK[1]+c_1\left (\frac {b \cot ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\lambda x)\right )}{a \lambda n+a \lambda }+y\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*cot(lambda*x)^n*diff(w(x,y),y) = c*cot(mu*x)^m+s*cot(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( \cot \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( \left ( -1+\cot \left ( \left ( -\int \!{\frac {b \left ( \cot \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \beta \right ) \cot \left ( {\frac {b\beta \,\int \! \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}}{a}} \right ) \right ) \left ( \cot \left ( \left ( -\int \!{\frac {b \left ( \cot \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \beta \right ) +\cot \left ( {\frac {b\beta \,\int \! \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}}{a}} \right ) \right ) ^{-1} \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \cot \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.6.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \cot ^n(\lambda y) w_y = c\cot ^m(\mu x)+s \cot ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Cot[lambda*y]^n*D[w[x, y], y] == c*Cot[mu*x]^m + s*Cot[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {\left (s \cot ^k(\beta K[1])+c \cot ^m\left (\frac {a \mu \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\cot ^2(\lambda y)\right ) \cot ^{1-n}(\lambda y)+b \lambda \mu x-b \lambda \mu n x-a \mu \cot ^{1-n}(\lambda K[1]) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\cot ^2(\lambda K[1])\right )}{b \lambda -b \lambda n}\right )\right ) \cot ^{-n}(\lambda K[1])}{b}dK[1]+c_1\left (\frac {\cot ^{1-n}(\lambda y) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\cot ^2(\lambda y)\right )}{\lambda (n-1)}-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*cot(lambda*y)^n*diff(w(x,y),y) = c*cot(mu*x)^m+s*cot(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =\int ^{y}\!{\frac { \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( s \left ( \cot \left ( \beta \,{\it \_b} \right ) \right ) ^{k}+ \left ( \left ( -1+\cot \left ( \mu \, \left ( -{\frac {a\int \! \left ( \cot \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \cot \left ( {\frac {a\mu \,\int \! \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}}{b}} \right ) \right ) \left ( \cot \left ( \mu \, \left ( -{\frac {a\int \! \left ( \cot \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) +\cot \left ( {\frac {a\mu \,\int \! \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}}{b}} \right ) \right ) ^{-1} \right ) ^{m}c \right ) }{d{\it \_b}}+{\it \_F1} \left ( -{\frac {a\int \! \left ( \cot \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \]
____________________________________________________________________________________