> [!NOTE] Theorem > Let $\frac{d}{dt} x(t)= f(x(t),t), \quad x(t_{0})=x_{0}\tag{1}$ be a [[Initial Value Problem for Scalar Ordinary Differential Equation|initial value problem for a first order scalar equation]] such that $f$ satisfies the [[Lipschitz Function|Lipschitz condition]]: for all $x_{1},x_{2}\in \mathbb{R},$ $|f(x_{1},t)-f(x_{2},t)|\leq L|x_{1}-x_{2}|$with some number $L>0.$ Then any two solutions for $(1)$ are equal. **Proof**: Suppose $x_{1}(t)$ and $x_{2}(t)$ are solutions to $(1).$ Let $w(t)=x_{1}(t)-x_{2}(t).$Then using chain rule, $\begin{align} \frac{d}{dt}(|w(t)|^{2}) &= 2w(t) \frac{d}{dt} (w(t)) \\ & = 2w(t) (f(x_{1}(t),t)-f(x_{2}(t),t)) \end{align}$Since $f$ satisfies the Lipschitz function, we have $\begin{align} \frac{d}{dt} (|w(t)|^{2}) & \leq 2 |w(t)| |f(x_{1}(t),t) - f(x_{2}(t),t)| \\ &\leq 2 |w(t)|L|x_{1}(t)-x_{2}(t)| \\ & \leq 2 L |w(t)|^{2} \end{align}$ Now [[Grönwall's Lemma]] asserts if nonnegative function $Z$ satisfies $\frac{d}{dt} Z(t)\leq c Z(t)$ then $Z(t)\leq Z(t_{0})e^{c(t-t_{0})}.$ Thus setting $Z(t)=|w(t)|^{2}$ and using the initial condition for $w,$ we see that $|w(t)|^{2} \leq |w(t_{0})|^{2}e^{2Lt} = 0.$As $|w(t)|^{2}\geq 0$ this necessarily means that $0=w(t)$ thus $x_{1}(t)=x_{2}(t).$