2.1006   ODE No. 1006

1. Problem in Latex
2. Mathematica input
3. Maple input

$$y^{″}(x)-y(x)=0$$ ✓ Mathematica : cpu = 0.00522438 (sec), leaf count = 20

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

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

$$\{y\left(x\right)={\mathrm{e}}^x\mathit{\_}\mathit{C}\mathit{1}+\mathit{\_}\mathit{C}\mathit{2}{\mathrm{e}}^{-x}\}$$

Hand solution

$$\begin{array}{}\text{(1)}& y^{\prime \prime }-y=0\end{array}$$ Let $y=e^{\lambda x}$, substitution in above gives$$\begin{array}{rl}{\lambda }^2{e}^{\lambda x}-e^{\lambda x}& =0\\ {\lambda }^2-1& =0\end{array}$$

Hence $\lambda =\pm 1$, therefore the solution is$$y_h=A{e}^x+B{e}^{-x}$$