ODE No. 1006

\[ y''(x)-y(x)=0 \]

✓ Mathematica : cpu = 0.0037038 (sec), leaf count = 20

DSolve[-y[x] + Derivative[2][y][x] == 0,y[x],x]

\[\left \{\left \{y(x)\to c_1 e^x+c_2 e^{-x}\right \}\right \}\]

✓ Maple : cpu = 0.008 (sec), leaf count = 15

dsolve(diff(diff(y(x),x),x)-y(x)=0,y(x))

\[y \left (x \right ) = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-x}\]

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

\[ y_{h}=Ae^{x}+Be^{-x}\]