Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 5 / c

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- Solution c

Solution

At the origin, combine the rational coefficients before taking the limit:

y (0) = \frac{1}{( 1 - a ^{2} ) ( 4 - a ^{2} )} - \frac{a}{3 ( 1 - a ^{2} )} + \frac{a}{6 ( 4 - a ^{2} )} = \frac{a + 3}{6 ( a + 1 ) ( a + 2 )} .

Equivalently

y (0) = L (3 L + 1) / [6 (L + 1) (2 L + 1)]

. The apparent poles at

L = 1

cancel, giving

L \to 1^{+} lim y (0) = \frac{1}{9} .

The entire limiting profile is also finite:

lim_{L \to 1^{+}} y (x) = \frac{( 1 + 3∣ x ∣ ) e ^{- ∣ x ∣} + e ^{- 2∣ x ∣}}{18},

which independently gives the same central value.

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