Linearization of Differential Equation

Question

Find a linearization of the differential equation for $x$ near $0$.

$$x''(t) + x(t) e^{0.05x} = 0$$

Not sure what to do here. My book isn't any help either..

Any help would be appreciated :)

Answer 1

When $x$ is small, $e^{0.05x} \approx 1$, therefore we can make the approximation

$$ x'' + x = 0 $$

Answer 2

Let $y=x’$. Then $y’=x’’=-x e^{ax}$ where $a=0.05$. Now you have a 2 dimensional dynamical system that can be linearised about x, y and written in matrix form as the linear system $$\left( \begin{array}{r}y’ \\ x’ \end{array} \right)= \left[ \begin{array}{rr} 0 & -e^{ax}(1+ax)\\ 1 & 0 \end{array} \right] \left( \begin{array}{r}y \\ x \end{array} \right)$$