Show that there is a unique solution to the equation \begin{equation} \frac{df}{dx}=(f(x)+x)x \tag{*}\end{equation} for $0 \le x \le 1$ and $f(0)=0$.
Clearly, $(*)$ is a first order linear ordinary differential equation and so, by the Picard-Lindelöf theorem, there should exist a unique solution. Nevertheless, I am not able to use this result as it stands since it is not part of the course I am currently reading. Instead, I was thinking of using the Contraction Principle. However, I am not entirely sure as to how I should accomplish this, any suggestions?
As remarked in the comment, there is another (way easier) way to solve the uniqueness. If you really want to use contraction mapping theorem:
Let $\mathscr C = \{ f \in C[0,1]: f(0) = 0\}$. $\mathscr C$ is a Banach space with norm
$$||f|| = \sup_{x\in [0,1]} |f(x)|.$$
Define $\Phi : C\to C$ by
$$\Phi(f) (x) = \int_0^x (f(s) + s)s ds . $$
Then for any $f, g\in \mathscr C$ $$|| \Phi (f)- \Phi(g)|| \leq \int_0^x s|f(s) - g(s)|ds \leq ||f - g||\int_0^x s ds \leq \frac 12 ||f-g||$$
Thus $\Phi$ is a contraction and so there is a unique $f\in \mathscr C$ so that
$$f(x) = \Phi(f) (x)= \int_0^x (f(s) + s)sds \Rightarrow \frac{df}{dx} (x)= (f(x) + x)x.$$