Ordinary differential equation/Separable variables/y' is g(t)y^2/Fact/Proof/Exercise

Prove that a differential equation of the shape

${}y'=g(t)\cdot y^{2}\,$

with a continuous function

$g\colon \mathbb {R} \longrightarrow \mathbb {R} ,t\longmapsto g(t),$

on an interval ${}I'$ has the solution

${}y(t)=-{\frac {1}{G(t)}}\,,$

where ${}G$ is an antiderivative of ${}g$ such that

${}G(I')\subseteq \mathbb {R} _{+}$.

Create a solution