2.7.5.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 30

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-a^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y = c_1 \,{\mathrm e}^{a x}+c_2 \,{\mathrm e}^{-a x}+c_3 \sin \left (a x \right )+c_4 \cos \left (a x \right ) \end{align*}

Maple trace

Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful