Solved as higher order constant coeﬀ ode

The characteristic equation is

\[ -a^{4}+\lambda ^{4} = 0 \]

The roots of the above equation are

\begin{align*} \lambda _1 &= a\\ \lambda _2 &= -a\\ \lambda _3 &= i a\\ \lambda _4 &= -i a \end{align*}

Therefore the homogeneous solution is

\[ y_h(x)={\mathrm e}^{i a x} c_1 +{\mathrm e}^{-i a x} c_2 +{\mathrm e}^{a x} c_3 +{\mathrm e}^{-a x} c_4 \]

The fundamental set of solutions for the homogeneous solution are the following

\begin{align*} y_1 &= {\mathrm e}^{i a x}\\ y_2 &= {\mathrm e}^{-i a x}\\ y_3 &= {\mathrm e}^{a x}\\ y_4 &= {\mathrm e}^{-a x} \end{align*}