 Solution

ID: 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 (- α) = 0

. On the left,

y (x) = \int_{- α}^{x} (1 - s)^{- 1/3} d s = \frac{3}{2} [(1 + α)^{2/3} - (1 - x)^{2/3}] .

Its value at the interface is

y (0) = \frac{3}{2} [(1 + α)^{2/3} - 1]

. On the right, integrate

(1 + x)^{- 1/2}

from the interface and enforce continuity. Therefore

y (x) = ⎩ ⎨ ⎧ \frac{3}{2} [(1 + α)^{2/3} - (1 - x)^{2/3}], \frac{3}{2} [(1 + α)^{2/3} - 1] + 2 [1 + x - 1], - α \leq x < 0, 0 \leq x \leq α .

Both

y

and

y^{'}

are continuous at zero. The ray steepens as it approaches the interface from the left and flattens as it leaves to the right.

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