Problem

乙、計算、證明題：共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

1. Suppose that $x$ and $y$ are related by the equation $x=\displaystyle\int_0^y \frac{1}{\sqrt{1+4t^2}}\,dt$. Show that $\dfrac{d^2y}{dx^2}$ is proportional to $y$ and find the constant of proportionality.