ODE
\[ y'''(x)+y''(x) (2 \cot (x)+\csc (x))-y'(x)=\cot (x) \] ODE Classification
[[_3rd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.680945 (sec), leaf count = 56
\[\left \{\left \{y(x)\to -\frac {c_2 x}{\sqrt {2}}-\frac {\cot \left (\frac {x}{2}\right ) \left (c_2 \log (2 (\cos (x)+1))+2 c_1\right )}{\sqrt {2}}+c_3+\cot \left (\frac {x}{2}\right ) \sin ^{-1}(\cos (x))\right \}\right \}\]
Maple ✓
cpu = 28.87 (sec), leaf count = 276
\[ \left \{ y \left ( x \right ) ={\frac {-{\it \_C1}\,\pi \, \left ( {\it csgn} \left ( i{{\rm e}^{-ix}} \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) \right ) ^{3}+{\it \_C1}\,\pi \, \left ( {\it csgn} \left ( i{{\rm e}^{-ix}} \right ) +{\it csgn} \left ( i \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) \right ) \left ( {\it csgn} \left ( i{{\rm e}^{-ix}} \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) \right ) ^{2}-{\it \_C1}\,\pi \,{\it csgn} \left ( i \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) {\it csgn} \left ( i{{\rm e}^{-ix}} \right ) {\it csgn} \left ( i{{\rm e}^{-ix}} \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) -{\it \_C1}\,\pi \, \left ( {\it csgn} \left ( i \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) \right ) ^{3}+2\,{\it \_C1}\,\pi \,{\it csgn} \left ( i+i{{\rm e}^{ix}} \right ) \left ( {\it csgn} \left ( i \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) \right ) ^{2}-{\it \_C1}\,\pi \, \left ( {\it csgn} \left ( i+i{{\rm e}^{ix}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ( {{\rm e}^{ix}}+1 \right ) ^{2} \right ) -2\,i{\it \_C1}\, \left ( {{\rm e}^{ix}}+1 \right ) \ln \left ( {{\rm e}^{ix}}+1 \right ) +2\,i{\it \_C1}\,\ln \left ( {{\rm e}^{ix}} \right ) + \left ( -ix-2\,{\it \_C1}\,x+{\it \_C3} \right ) {{\rm e}^{ix}}+2\,i\ln \left ( 2 \right ) {\it \_C1}-ix-2\,i{\it \_C2}+2\,{\it \_C1}\,x-{\it \_C3}}{{{\rm e}^{ix}}-1}} \right \} \] Mathematica raw input
DSolve[-y'[x] + (2*Cot[x] + Csc[x])*y''[x] + y'''[x] == Cot[x],y[x],x]
Mathematica raw output
{{y[x] -> -((x*C[2])/Sqrt[2]) + C[3] + ArcSin[Cos[x]]*Cot[x/2] - (Cot[x/2]*(2*C[1] + C[2]*Log[2*(1 + Cos[x])]))/Sqrt[2]}}
Maple raw input
dsolve(diff(diff(diff(y(x),x),x),x)+(2*cot(x)+csc(x))*diff(diff(y(x),x),x)-diff(y(x),x) = cot(x), y(x),'implicit')
Maple raw output
y(x) = (-_C1*Pi*csgn(I*exp(-I*x)*(exp(I*x)+1)^2)^3+_C1*Pi*(csgn(I*exp(-I*x))+csgn(I*(exp(I*x)+1)^2))*csgn(I*exp(-I*x)*(exp(I*x)+1)^2)^2-_C1*Pi*csgn(I*(exp(I*x)+1)^2)*csgn(I*exp(-I*x))*csgn(I*exp(-I*x)*(exp(I*x)+1)^2)-_C1*Pi*csgn(I*(exp(I*x)+1)^2)^3+2*_C1*Pi*csgn(I+I*exp(I*x))*csgn(I*(exp(I*x)+1)^2)^2-_C1*Pi*csgn(I+I*exp(I*x))^2*csgn(I*(exp(I*x)+1)^2)-2*I*_C1*(exp(I*x)+1)*ln(exp(I*x)+1)+2*I*_C1*ln(exp(I*x))+(-I*x-2*_C1*x+_C3)*exp(I*x)+2*I*ln(2)*_C1-I*x-2*I*_C2+2*_C1*x-_C3)/(exp(I*x)-1)