Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 9 / b / ii / Solution

Integrate the slope in part (i), using $y(-\alpha )=0$. On the left,Its value at the interface is $y(0)=\frac{3}{2}[(1+\alpha {)}^{2/3}-1]$. On the right, integrate $(1+x{)}^{-1/2}$ from the interface and enforce continuity. ThereforeBoth $y$ and $y^{\prime }$ are continuous at zero. The ray steepens as it approaches the interface from the left and flattens as it leaves to the right.