General Solution to System of Differential equations

Question

I'm asked to find the general solution of this non-linear system of differential equations. \begin{align*} \begin{cases} x'(t) &= y(t) \\ y'(t) &= x(t) - x^2(t) \end{cases} \end{align*} I'm in an introductory course so I've only seen some cualitative aspects of these equations but no method to solve them, though I saw I could derive $x'(t)$ again and change the problem to solving: \begin{align*} x''(t) = x(t) - x^2(t) \end{align*} Then by setting $z(t)=x'(t)$ I could change variable to get: \begin{align*} z(t)\frac{dz}{dx} = x(t)-x^2(t) \end{align*} Which is separable so $z(t)=\sqrt{x^2(t) - \frac{2}{3}x^3(t) + C}$. However the integral solution of this doesn't appear to have any nice solution even with tracendental functions, so my question is if there is another way to get the solution, which yields a more solvable integral? I don't think my professor has any problem if the solution isn't given explicitly, but I don't think I can just leave it as an integral.