\[ -a^2 y'(x)+2 a^2 y(x)+y^{(3)}(x)-2 y''(x)-\sinh (x)=0 \] ✓ Mathematica : cpu = 0.0816891 (sec), leaf count = 95
DSolve[-Sinh[x] + 2*a^2*y[x] - a^2*Derivative[1][y][x] - 2*Derivative[2][y][x] + Derivative[3][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {e^{-x} \left (3 a^2 e^{2 x}-a^2-3 e^{2 x}-12 e^x \sinh (x)-6 e^x \cosh (x)+1\right )}{6 (a-2) (a+2) \left (a^2-1\right )}+c_1 e^{-a x}+c_3 e^{a x}+c_2 e^{2 x}\right \}\right \}\] ✓ Maple : cpu = 0.102 (sec), leaf count = 113
dsolve(diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-a^2*diff(y(x),x)+2*a^2*y(x)-sinh(x)=0,y(x))
\[y \left (x \right ) = \frac {\left (6 a^{4} c_{3}-30 a^{2} c_{3}+24 c_{3}\right ) {\mathrm e}^{-a x}+6 \left (a -1\right ) \left (c_{1} a^{2}+\frac {\sinh \left (3 x \right )}{6}-4 c_{1}-\frac {\cosh \left (3 x \right )}{6}\right ) \left (a +1\right ) {\mathrm e}^{2 x}+\left (6 a^{4} c_{2}-30 a^{2} c_{2}+24 c_{2}\right ) {\mathrm e}^{a x}+3 \,{\mathrm e}^{x} a^{2}-12 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-x}}{6 a^{4}-30 a^{2}+24}\]