Example of linear first order differential equation, with non unique solution.

Question

I'm was stubling on the question of uniqueness of differential equations. I have a differential equation of the form $ \frac{d}{dt}y(t)= f(t)y(t), y(t_0)=p_0$ ie. a linear, first order differential equation and it seems to be the case, that one of the ways to determine uniqueness of the solutionfunction is to require continuity on a open interval of f(t). I couldn't find example which were not unique, could only find some where the diff. eq. was nonlinear. My question, does such exist and can you give me an example?

Answer 1

The condition of unicity and existence of a first order ODE $y'=g(t,y)$ is that $g(t,y)$ is Lipschitz continuous in $y$ (it is the statement of the Picard-Lindelof theorem).

By the definition of Lipschitz continuity, $f(t) y(t)$ is Lipschitz continuous if $\exists K$ such that $\forall y$ $$ |f y_1 - f y_2| \leq K |y_1 - y_2| $$ which is obviously satisfied if $K \geq f$ (see that $K$ is constant in relation to $y$).

Therefore, any ODE in the form $y'=f(t)y(t)$ has a unique solution.

Answer 2

By the Picard–Lindelöf theorem, the solution to your problem is always unique.