Let $g$ be the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ with radius of convergence $R \geqslant \pi/2$. Show $$\forall x \in I, \quad 2g^{\prime}(x) = g(x)^2 + 1.$$
Let $g$ be the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ with radius of convergence $R \geqslant \pi/2$. Show
$$\forall x \in I, \quad 2g^{\prime}(x) = g(x)^2 + 1.$$