ODE
\[ -\left (\left (a^2+1\right ) y(x)\right )+y''(x)-2 \tan (x) y'(x)=\sin (x) \] ODE Classification
[[_2nd_order, _linear, _nonhomogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.249751 (sec), leaf count = 46
\[\left \{\left \{y(x)\to \frac {1}{2} \sec (x) \left (-\frac {\sin (2 x)}{a^2+4}+2 c_1 e^{-a x}+\frac {c_2 e^{a x}}{a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.547 (sec), leaf count = 36
\[\left [y \left (x \right ) = \frac {\sinh \left (a x \right ) \textit {\_C2}}{\cos \left (x \right )}+\frac {\cosh \left (a x \right ) \textit {\_C1}}{\cos \left (x \right )}-\frac {\sin \left (x \right )}{a^{2}+4}\right ]\] Mathematica raw input
DSolve[-((1 + a^2)*y[x]) - 2*Tan[x]*y'[x] + y''[x] == Sin[x],y[x],x]
Mathematica raw output
{{y[x] -> (Sec[x]*((2*C[1])/E^(a*x) + (E^(a*x)*C[2])/a - Sin[2*x]/(4 + a^2)))/2}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-2*diff(y(x),x)*tan(x)-(a^2+1)*y(x) = sin(x), y(x))
Maple raw output
[y(x) = 1/cos(x)*sinh(a*x)*_C2+1/cos(x)*cosh(a*x)*_C1-sin(x)/(a^2+4)]