\[ y''(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.0036825 (sec), leaf count = 20
\[\left \{\left \{y(x)\to c_1 e^x+c_2 e^{-x}\right \}\right \}\] ✓ Maple : cpu = 0.007 (sec), leaf count = 15
\[y \relax (x ) = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}\]
Hand solution
\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\[ y_{h}=Ae^{x}+Be^{-x}\]