Definition of the Theorem of Unique Solution Existence in Differential Equations

Let $R$ be a rectangular region in the $xy$-plane defined by $a\le x\le b$ and $c\le y\le d$ that contains point $(x_0,y_0)$.

If $f(x,y)$ and $\frac{\partial f}{\partial y}$ are continuous on $R$, then there exists…

• … some interval $I_0:(x_0-h,x_0+h),h>0$, contained in $[a,b]$,
• … a unique function $y(x)$ defined on $I_0$, that is a solution of the initial value problem (IVP):

$$\frac{dy}{dx}=f(x,y),y(x=x_0)=y_0$$