\[ a^2 \left (-y'(x)\right )-e^{2 a x} \sin ^2(x)+y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 1.01629 (sec), leaf count = 128
DSolve[-(E^(2*a*x)*Sin[x]^2) - a^2*Derivative[1][y][x] + Derivative[3][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {e^{-a x} \left (-9 \left (a^2-4\right ) a^4 e^{3 a x} \cos (2 x)-3 \left (11 a^2-4\right ) a^3 e^{3 a x} \sin (2 x)+\left (9 a^6+49 a^4+56 a^2+16\right ) \left (12 a^2 c_1 e^{2 a x}-12 a^2 c_2+e^{3 a x}\right )\right )}{12 a^3 \left (9 a^6+49 a^4+56 a^2+16\right )}+c_3\right \}\right \}\] ✓ Maple : cpu = 0.139 (sec), leaf count = 122
dsolve(diff(diff(diff(y(x),x),x),x)-a^2*diff(y(x),x)-exp(2*a*x)*sin(x)^2=0,y(x))
\[y \left (x \right ) = \frac {\left (\left (-9 a^{6}+36 a^{4}\right ) \cos \left (2 x \right )+\left (-33 a^{5}+12 a^{3}\right ) \sin \left (2 x \right )+9 a^{6}+49 a^{4}+56 a^{2}+16\right ) {\mathrm e}^{2 a x}+108 \left (a^{2}+4\right ) \left (a^{2}+\frac {4}{9}\right ) \left (a^{2}+1\right ) a^{2} \left (a c_{3}-{\mathrm e}^{-a x} c_{1}+{\mathrm e}^{a x} c_{2}\right )}{108 a^{9}+588 a^{7}+672 a^{5}+192 a^{3}}\]