\[ 2 f'(x) \left (y^{(3)}(x)-a^2 y'(x)\right )+f(x) \left (-2 a^2 y''(x)+a^4 y(x)+y^{(4)}(x)\right )=0 \] ✗ Mathematica : cpu = 0.163121 (sec), leaf count = 0 , could not solve
DSolve[2*Derivative[1][f][x]*(-(a^2*Derivative[1][y][x]) + Derivative[3][y][x]) + f[x]*(a^4*y[x] - 2*a^2*Derivative[2][y][x] + Derivative[4][y][x]) == 0, y[x], x]
✓ Maple : cpu = 0.124 (sec), leaf count = 67
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{ax}}+{\it \_C2}\,{{\rm e}^{-ax}}+{\it \_C3}\,{{\rm e}^{{\frac {x}{f} \left ( -{\it df}+\sqrt {{a}^{2}{f}^{2}+{{\it df}}^{2}} \right ) }}}+{\it \_C4}\,{{\rm e}^{-{\frac {x}{f} \left ( {\it df}+\sqrt {{a}^{2}{f}^{2}+{{\it df}}^{2}} \right ) }}} \right \} \]