# Solve the following non linear system of differential equations

Consider the non-linear system,

$$\left( \begin{array}{ccc} x_{1}' \\ x_{2}' \end{array} \right) = \left( \begin{array}{ccc} -x_{1} \\ x_{2}+x_{1}^2 \end{array} \right)$$

Find the solution of the initial value problem..I couldn't integrate the the equation to find $x_{2}$,since it involves two dependent variables $x_{1}$ and ${x_{2}}$.

Can someone help me for the same..

$x_1$ is not coupled to $x_2$. From the equation for $x_1'$, it's not hard to find that $$x_1(t) = e^{-t}x_1(0).$$
Can you substitute this into the equation for $x_2'$ and solve from there?