Initial value problem, open rectangle

Question

I have no idea how to come up with this.

Give an example of an initial value problem for which the open rectangle

$R = [ (x,y) : 0 < x < 4, -1

represents the largest region in the xy-plane where the hypotheses of Existence and Uniqueness Theorem are satisfied.

Answer 1

For a general first order ODE $F(x,y) = y'$ with initial value $y(x_0)= y_0$ to be guaranteed existence or uniqueness on a rectangular region $R$, it must be continuous on that region, and for uniqueness to hold, its derivative with respect to $y$ must also be continuous. Of course the point $(x_0, y_0)$ also needs to be contained in $R$.

To solve your problem, you must then do the following: specify an appropriate initial value in your given $R$. Define $F(x,y)$ such that $F(x,y)$ is continuous and has continuous derivative with respect to $y$ in the region $R$, but such that $F(x,y)$ hits 'walls' at the boundaries of $R$ where continuity fails. Big hint:

think about the function $\frac{1}{y-a}$, which is, among other things, continuous everywhere except at $y=a$.