# differential equations – A bound on a solution of an ODE, given some bounds on endpoints

Let $$V : (a,b) to mathbb{R}$$ be smooth, strictly increasing and $$V(a) = 0$$. Suppose that $$f : (a,b) to mathbb{R}$$ is smooth and satisfies $$f^{prime prime} (x) + V(x) f(x) = 0$$ on $$(a,b)$$. Can we then bound $$sup_{x in (a,b)} |f(x)|$$ in terms of $$f(a) , f(b) , f^{prime} (b)$$? I intentionally don’t put $$f^{prime} (a)$$ into the list.