3 Mathematica. Time used: 0.004 (sec). Leaf size: 53

ode=D[y[x],{x,4}]-a^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]

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