What are continous coefficients in an ODE?

Question

I have studied ODEs but today my teacher asked a question, that I was confused about:
If an ODE is given as a linear initial value problem in normalized form, where coefficients are continuous,then will the solution be be unique?
I don't understand by what my prof meant by continous coefficients in an ODE, and would greatly appreciate some help.

Answer 1

The ODE is linear with leading coefficient $1$ (which is the meaning of normalized), that is, $$ y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\ldots+a_0(x)y(x)=r(x). $$ The assumption is that all coefficients $a_k$ and right side $r$ are continuous functions. Then the equation is globally Lipschitz in every strip over a bounded $x$-interval, with the claimed consequences on existence and uniqueness.