Past exam of the differential equations 1 course of the University of Oxford / 2023 / 1 / b / ii

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- Solution ii

Solution

Put

B = ∣ b ∣ > 0

and

γ = 4 B / C_{1} < 1

. Initially

M_{0} = B < 2 B

. If

M_{n} < 2 B

, the preceding supremum norm estimate gives

M_{n + 1} \leq B + \frac{M _{n}^{2}}{C _{1}} < B + \frac{4 B ^{2}}{C _{1}} = B (1 + γ) < 2 B .

Hence mathematical induction gives

t \geq 0 sup ∣ y_{n} (t) ∣ < 2∣ b ∣ for every n \geq 0.

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