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

Fix any $t_0\ge 0$ and compare $y=R$ with $y=0$. The quotient required in a global Lipschitz condition in the dependent variable isConsequently,It is nevertheless locally Lipschitz in $y$: for $|y|,|v|\le R$,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.

Write $A(t)={\int }_0^{t}a(s)ds$. Multiplying the ordinary differential equation by the integrating factor $e^{A(t)}$ and using the product rule givesIntegration from $0$ to $t$ therefore yields the Volterra integral equationConversely, a continuous function satisfying this integral equation becomes continuously differentiable after multiplication by $e^{A(t)}$, by the fundamental theorem of calculus, and satisfies the original initial value problem.