 Solution

ID: past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2023/1/a/i/solution

Fix any

t_{0} \geq 0

and compare

y = R

with

y = 0

. The quotient required in a global Lipschitz condition in the dependent variable is

\frac{∣ f ( t _{0} , R ) - f ( t _{0} , 0 ) ∣}{∣ R - 0∣} = ∣ R - a (t_{0}) ∣ ⟶ \infty (R ⟶ \infty) .

Consequently,

f is not globally Lipschitz in y .

It is nevertheless locally Lipschitz in

y

: for

∣ y ∣, ∣ v ∣ \leq R

∣ f (t, y) - f (t, v) ∣ = ∣ y - v ∣ ∣ y + v - a (t) ∣ \leq (2 R + C_{2}) ∣ y - v ∣.

Thus the Picard-Lindelöf theorem gives local existence and uniqueness for the initial value problem; the remaining work establishes existence on the entire half-line for the specified small data.

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