\[ \boxed { f \left ( {\it d4y} \left ( x \right ) -2\,{a}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +{a}^{4}y \left ( x \right ) \right ) +2\,{\it df}\, \left ( {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) -{a}^{2}{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 0.246031 (sec), leaf count = 50 \[ \text {DSolve}\left [2 f'(x) \left (y^{(3)}(x)-a^2 y'(x)\right )+f(x) \left (a^4 y(x)-2 a^2 y''(x)+y^{(4)}(x)\right )=0,y(x),x\right ] \]
Maple: cpu = 0.016 (sec), leaf count = 67 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{ax}}+{\it \_C2}\,{ {\rm e}^{-ax}}+{\it \_C3}\,{{\rm e}^{{\frac {x}{f} \left ( -{\it df}+ \sqrt {{a}^{2}{f}^{2}+{{\it df}}^{2}} \right ) }}}+{\it \_C4}\,{{\rm e} ^{-{\frac {x}{f} \left ( {\it df}+\sqrt {{a}^{2}{f}^{2}+{{\it df}}^{2}} \right ) }}} \right \} \]