Linear inhomogeneous differential equation/Superposition principle/Exercise

Let ${}I\subseteq \mathbb {R}$ be a real interval and let

$g,h_{1},h_{2}\colon I\longrightarrow \mathbb {R}$

be functions. Let ${}y_{1}$ be a solution to the differential equation ${}y'=g(t)y+h_{1}(t)$ and let ${}y_{2}$ be a solution to the differential equation ${}y'=g(t)y+h_{2}(t)$. Prove that ${}y_{1}+y_{2}$ is a solution to the differential equation

${}y'=g(t)y+h_{1}(t)+h_{2}(t)\,.$

Create a solution