The exercise is:
Approximate the solutions of $x'' + x + ε*x^3 = 0$, the initial value is $x(0) = 1$ and $x'(0) = 0$ up to order one. (Hint: It is not necessary to convert this second order equation to a first order system.)
To solve this, we have to use Regular perturbation theory: we have to get it with the equation: $\varphi(t, \epsilon) = \varphi_{0}(t) + \varphi_{1}(t)\epsilon + o(\epsilon)$ where $\varphi$ is the solution of the ode and $\epsilon$ is the perturbation.
$\varphi_{0}(t) = \varphi(t, 0)$ is given as the solution of the unperturbed equation where $\epsilon = 0$
$$\varphi_{1}(t) = \frac{\partial \varphi(t,\epsilon)}{\partial \epsilon} \bigg|_{\epsilon=0}$$
how can I compute $\varphi_{1}(t)$? or is this the wrong approach?
First you solve the unperturbed equation $x''+x=0$ with the given initial conditions to get $\phi_0(t)$. Can you do that? Then you use that to approximate the cubic term. You need to find the solution to $\phi_1(t)''+\phi_1(t)+\epsilon \phi_0(t)^3=0$. You are imagining that the effect of the cubic term is small so the solution will be approximately $\phi_0(t)$. That gives you a nice second order differential equation to solve.