# Recitation 4

### Existence and uniqueness theorems for 1st-order ODE

The general 1st-order initial value problem (IVP) is $$\begin{equation}\tag{*}y’=F(x,y), y(x_0) = y_0.\end{equation}$$ We are interested in the following questions:

1. Under what conditions can we be sure that a solution to (*) exists?
2. Under what conditions can we be sure that there is a unique solution to (*)?

Theorem (Existence and uniqueness). Suppose that F(x,y) is a continuous function defined in some region $$R = (x_0-\delta, x_0+\delta)\times (y_0-\epsilon, y_0+\epsilon)$$ containing the point (x_0, y_0). Then there exists \delta_1 > 0 so that a solution y=f(x) to (*) is defined for x\in (x_0-\delta_1, x_0+\delta_1). Suppose, furthermore, that \frac{\partial F}{\partial y}(x,y) is a continuous function defined on R. Then there exists \delta_2 > 0 so that the solution is the unique solution to (*) for x\in (x_0-\delta_2, x_0 + \delta_2).

Example 1 Consider the IVP y'=x-y+1, y(1)=2. In this case, both the F(x,y)=x-y+1 and \frac{\partial F}{\partial y}(x,y)=-1 are defined and continuous at all points. The theorem guarantees that a solution to the ODE uniquely exists in some open interval centered at 1. In fact, an explicit solution to this equation is y(x) = x+e^{1-x}. This solution exists for all x.