\[ y''(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.00472456 (sec), leaf count = 20
\[\left \{\left \{y(x)\to c_1 e^x+c_2 e^{-x}\right \}\right \}\]
✓ Maple : cpu = 0.013 (sec), leaf count = 15
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{x}}+{\it \_C2}\,{{\rm e}^{-x}} \right \} \]
\begin {equation} y^{\prime \prime }-y=0\tag {1} \end {equation} Let \(y=e^{\lambda x}\), substitution in above gives\begin {align*} \lambda ^{2}e^{\lambda x}-e^{\lambda x} & =0\\ \lambda ^{2}-1 & =0 \end {align*}
Hence \(\lambda =\pm 1\), therefore the solution is\[ y_{h}=Ae^{x}+Be^{-x}\]