Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 7A (Methods)

Table of contents
- a 7A
  - Solution a
- b 7A
  - Solution b

a

Solution

Write

G (x; ξ) = e^{2 x} H (x; ξ)

. Since

(D - 2)^{2} (e^{2 x} H) = e^{2 x} H^{''}

, the Green function must have

H (0; ξ) = H (1; ξ) = 0

, be continuous at

x = ξ

, and satisfy the jump condition for an impulse

H_{x} (ξ^{+}; ξ) - H_{x} (ξ^{-}; ξ) = e^{- 2 ξ}

. Solving the two linear pieces gives

G (x; ξ) = e^{2 (x - ξ)} {x (ξ - 1), ξ (x - 1), 0 \leq x \leq ξ, ξ \leq x \leq 1.

b

Solution

The Green-function representation gives

y (x) = \int_{0}^{1} G (x; ξ) e^{2 ξ} d ξ = e^{2 x} [(x - 1) \int_{0}^{x} ξ d ξ + x \int_{x}^{1} (ξ - 1) d ξ] .

The bracket simplifies to

x (x - 1) /2

, so

y (x) = \frac{1}{2} x (x - 1) e^{2 x} .

It vanishes at both endpoints, and applying

(D - 2)^{2}

returns

e^{2 x}

.

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