Why is continuity Required for Existence and uniqueness theorem
Recently i am studying "Existence and uniqueness theorem" first order differential
equation
ie my IVP is $\frac{dy}{dx}=f(x,y), y(x_0)=y_0$
and here is $f(x,y) $ is continuity in both $x,y$ and so from this we studied "Uniqueness" of the second order by converting system of equation and hence we studied "Uniqueness" of $n^{th}$ order differential
and my first question is why need continuous always what if i remove continuity
and for non-homogeneous differential equation
id $y''+ay'+by=f(x)$
my second question is for what condition of $f(x)$ we can apply Uniqueness theorem
I am sorry i dont know this is good question or not but i got this questions in my mind
Best Answer
Look up Peano's existence theorem and the Picard-Lindelof theorem. You need "for all $x$, the function $y \mapsto f(x,y)$ is Lipschitz continuous" to get uniqueness. There's a counterexample at the bottom of the relevant wikipedia page.