Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
The purpose of this question is to prove that there exist $(\ell, M_{0}) \in \mathbb{R}_{+}^{2}$ such that $$\forall x > 0, \left||f(x)|^{2} - 1\right| \leq \frac{M_{0}}{x}$$
(a) Show that $$\forall x \in \mathbb{R}, \Im\left(f^{\prime}(x)\overline{f(x)}\right) + \frac{x}{4}|f(x)|^{2} - \frac{1}{4}\int_{0}^{x}|f(t)|^{2}\,dt = 0$$
(b) Show that $$\forall x > 0, \frac{d}{dx}\left(\frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt\right) = -\frac{4}{x^{2}}\Im\left(f^{\prime}(x)\overline{f(x)}\right)$$
(c) Deduce that there exists $\ell \in \mathbb{R}_{+}$ such that $$\lim_{x \rightarrow +\infty} \frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt = \ell$$
(d) Show that there exists $M \in \mathbb{R}_{+}$ such that $$\forall x > 0, \left|\frac{1}{x}\int_{0}^{x}|f(t)|^{2}\,dt - \ell\right| \leq \frac{M}{x}$$
(e) Conclude.