ODE
\[ y''(x)+y'(x) (\cot (x)+\csc (x))=a \csc (x)+1 \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.520098 (sec), leaf count = 77
\[\left \{\left \{y(x)\to \frac {2 i a \left (1+e^{i x}\right ) \log \left (1+e^{i x}\right )+e^{i x} (2 a x-i x+1+2 c_2)-i x-1+2 i c_1}{-1+e^{i x}}\right \}\right \}\]
Maple ✓
cpu = 3.655 (sec), leaf count = 368
\[\left [y \left (x \right ) = -\frac {x}{\tan \left (x \right )}+\ln \left (\tan \left (x \right )\right )-\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}-\frac {\textit {\_C1}}{\tan \left (x \right )}-\frac {\ln \left (-1+\cos \left (2 x \right )\right )}{2}-\frac {a \ln \left (2\right )}{\tan \left (x \right )}-\frac {x}{\sin \left (x \right )}-\frac {\textit {\_C1}}{\sin \left (x \right )}-\frac {a \ln \left (2\right )}{\sin \left (x \right )}+\frac {2 i a \ln \left (\frac {1-\frac {1}{1+{\mathrm e}^{i x}}}{1+{\mathrm e}^{i x}}\right )}{\left (\frac {2}{1+{\mathrm e}^{i x}}-1\right ) \left (1+{\mathrm e}^{i x}\right )}-\frac {2 i a \ln \left (\frac {1-\frac {1}{1+{\mathrm e}^{i x}}}{1+{\mathrm e}^{i x}}\right )}{\left (\frac {2}{1+{\mathrm e}^{i x}}-1\right ) \left (1+{\mathrm e}^{i x}\right )^{2}}-\frac {2 i a \,{\mathrm e}^{2 i x} \ln \left (\frac {{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}+2 \,{\mathrm e}^{i x}+1}\right )}{{\mathrm e}^{2 i x}-1}-2 i a \ln \left (1+{\mathrm e}^{i x}\right )-\frac {i a \ln \left (2\right )}{1+{\mathrm e}^{i x}}-\frac {2 i a}{\left (\frac {2}{1+{\mathrm e}^{i x}}-1\right ) \left (1+{\mathrm e}^{i x}\right )}+\frac {i a \ln \left (2\right )}{\frac {4}{1+{\mathrm e}^{i x}}-2}-\frac {2 i a \ln \left (2\right )}{{\mathrm e}^{2 i x}-1}-\frac {2 i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}+\frac {4 i a}{\left (\frac {2}{1+{\mathrm e}^{i x}}-1\right ) \left (1+{\mathrm e}^{i x}\right )^{2}}+\frac {2 i a}{{\mathrm e}^{2 i x}-1}+\textit {\_C2}\right ]\] Mathematica raw input
DSolve[(Cot[x] + Csc[x])*y'[x] + y''[x] == 1 + a*Csc[x],y[x],x]
Mathematica raw output
{{y[x] -> (-1 - I*x + (2*I)*C[1] + E^(I*x)*(1 - I*x + 2*a*x + 2*C[2]) + (2*I)*a*(1 + E^(I*x))*Log[1 + E^(I*x)])/(-1 + E^(I*x))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(cot(x)+csc(x))*diff(y(x),x) = 1+a*csc(x), y(x))
Maple raw output
[y(x) = -x/tan(x)+ln(tan(x))-1/2*ln(1+tan(x)^2)-_C1/tan(x)-1/2*ln(-1+cos(2*x))-a*ln(2)/tan(x)-1/sin(x)*x-_C1/sin(x)-a*ln(2)/sin(x)+2*I*a/(2/(1+exp(I*x))-1)/(1+exp(I*x))*ln(1/(1+exp(I*x))*(1-1/(1+exp(I*x))))-2*I*a/(2/(1+exp(I*x))-1)*ln(1/(1+exp(I*x))*(1-1/(1+exp(I*x))))/(1+exp(I*x))^2-2*I*a/(exp(2*I*x)-1)*exp(2*I*x)*ln(exp(I*x)/(exp(2*I*x)+2*exp(I*x)+1))-2*I*a*ln(1+exp(I*x))-I*a/(1+exp(I*x))*ln(2)-2*I*a/(2/(1+exp(I*x))-1)/(1+exp(I*x))+1/2*I*a*ln(2)/(2/(1+exp(I*x))-1)-2*I*a*ln(2)/(exp(2*I*x)-1)-2*I*a/(exp(2*I*x)-1)*exp(I*x)+4*I*a/(2/(1+exp(I*x))-1)/(1+exp(I*x))^2+2*I*a/(exp(2*I*x)-1)+_C2]