Existence and Uniqueness

[Anonymous No.2651711]

consider an initial value problem
<span class="math"> \frac{dy}{dx} = f(x,y),~y(x_0)=y_0 [/spoiler]
if f(x,y) is continuous in some rectangle
<span class="math"> R= \left \{ (x,y)| \left | x-x_0 \right | <a , \left | y-y_0 \right | <b\right \} [/spoiler]
then the initial value problem has at least one solution.


What the fuck does this mean and how do I use it to solve a question?

If I'm given a first order ODE do I find y and then use it somehow, or rearrange it to get f(x,y) on it's own? None of this makes any sense to me.