RO-NMO 2014 Problem 122

Let $I,J$ be two intervals, $\varphi :J\longrightarrow\mathbb{R}$ be a continuous function whose image doesn't contain $0,$ and $f,g:I\longrightarrow J$ be two differentiable functions such that $f'=\varphi\circ f,g'=\varphi\circ g$ and such that the image of $f-g$ contains $0.$ Show that $f$ and $g$ are the same function.