Specific Theorem For Uniqueness of Solution

Question

I'm currently studying Differential Equations for independent study. I've across the claim that given some ordinary differential equation (initial value problem) $f(t,y)$, if $f$ is continuous in some domain $D$ of the $yt$ plane, and the partial derivative $\frac{\partial f}{\partial y}$ is also continuous on that same domain, then we know that there exists some unique solution to the differential equation. Is this claim actually true? If so, is there a name or theorem associated with this result?

Answer 1

The theorem states that:

Let the functions $f$ and $\frac{\partial f}{\partial y}$ be continuous in some rectangle $\alpha<t<\beta$, $\gamma<y<\delta$ containing the point $(t_{0},y_{0})$. Then, in some interval $t_{0}-h<t<t_{0}+h$ contained in $\alpha<t<\beta$, there's aunique solution $y=\phi(t)$ of the initial value problem $$y'=f(t,y), \quad y(y_{0})=y_{0}$$

Note that is important the initial condition for the problem. Here we note that the conditions stated in theorem are sufficients to guarentee the existence of a unique solution of the initial value problem in some interval $t_{0}-h<t<t_{0}+h$, but they're not necessary. That's the conclusion remains true under slightly weaker hypotheses about the function $f$. In fact, the existence of a solution but not its uniqueness can be established on the basis of the continuity of $f$ alone.

An important geometrical consequence of the uniqueness part of theorem is that the graph of two solutions can't intersect each other. Otherwise, there would be two solutions that satisfy the initial condition corresponding to the point of intersection, in contradiction to theorem.

Finally, you can read Picard–Lindelöf theorem for more information.

For your self-education I suggest you also read this text online which contains theorems and practical examples. Good luck!