 Solution

ID: past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2023/1/b/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 .

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

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