Show that there's an interval $I \subseteq \mathbb{R}$ with $3 \in I$, so that the initial value problem $$ x'(t) = (x(t) - t)e^{t^2+x(t)} \text{ with } x(3) = 7$$ has a unique solution.
So far I have only found one particular way of solving IVPs (which is finding functions $P(t)$ and $f(t)$ so that $\frac{dy}{dt} + P(t)y = f(t), y(t_0) = y_0$). Are there any other ways that would be easier to apply to this IVP that I've written above? I'm assuming Picard-Lindelof, but I have yet to see a comprehendable approach to this problem. Thank you for your input.