Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 3B (Methods)

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- Solution 3B

Solution

Let

r_{1}, r_{2}

be the roots of

α r^{2} + β r + γ = 0

. The causal Green function is

G (t, τ) = H (t - τ) g (t - τ)

, where

g (s) = \frac{e ^{r_{1} s} - e ^{r_{2} s}}{α ( r _{1} - r _{2} )}

for distinct roots, and

g (s) = s e^{rs} / α

for a repeated root. These formulas enforce

g (0) = 0

and

α g^{'} (0) = 1

, hence the required delta jump.

For the oscillator,

y (t) = \int_{0}^{t} \frac{s i n ( ω ( t - τ ))}{ω} sin (λ τ) d τ .

If

λ \neq = ω

, this is

y (t) = \frac{s i n ( λ t ) - ( λ / ω ) s i n ( ω t )}{ω ^{2} - λ ^{2}} .

λ = ω

, the limiting expression is

y (t) = \frac{sin ( ω t ) - ω t cos ( ω t )}{2 ω ^{2}} .

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