📂Odinary Differential Equations

Picard's Theorem

Buildup¹

Consider the following ODE system.

$$\begin{equation} \begin{aligned} x_{1}^{\prime}(t) =&\ F_{1}(t,x_{1},x_{2},\cdots,x_{n}) \\ x_{2}^{\prime}(t) =&\ F_{2}(t,x_{1},x_{2},\cdots,x_{n}) \\ \vdots & \\ x_{n}^{\prime}(t) =&\ F_{n}(t,x_{1},x_{2},\cdots,x_{n}) \end{aligned} \end{equation}$$

Assume the values of $x_{i}$ are as follows when $t=t_{0}$.

$$\begin{equation} x_{1}(t_{0}) = x_{1}^{0}, x_{2}(t_{0}) = x_{2}^{0}, \dots, x_{n}(t_{0}) = x_{n}^{0} \end{equation}$$

Combining $(1)$ and $(2)$ into an initial value problem of a system of first-order differential equations, and finding the solution $x_{1} = \phi_{1}(t), x_{2} = \phi_{2}(t), \dots, x_{n} = \phi_{n}(t)$ to this problem is referred to as solving the initial value problem.

Theorem

Let’s assume there are $n$ functions $F_{1}, \dots, F_{n}$ and $n^{2}$ first derivatives $\dfrac{\partial F_{1}}{\partial x_{1}}, \dots, \dfrac{\partial F_{1}}{\partial x_{n}}, \dots, \dfrac{\partial F_{n}}{\partial x_{1}}, \dots, \dfrac{\partial F_{n}}{\partial x_{n}}$, all continuous in some domain $R = \left\{ (t, x_{1},\dots, x_{n}) : \alpha \lt t \lt \beta, \alpha_{1} \lt x_{1} \lt \beta_{1}, \dots, \alpha_{n} \lt x_{n} \lt \beta_{n} \right\}$. Suppose point $\left( t_{0}, x_{1}^{0}, \dots, x_{n}^{0} \right)$ is a point in $R$.

Then, there exists a unique solution $x_{1} = \phi_{1}(t), x_{2} = \phi_{2}(t), \dots, x_{n} = \phi_{n}(t)$ to the initial value problem $(1), (2)$ in some interval $\left| t - t_{0} \right| \lt h$.

Explanation

The statement that a solution to the initial value problem of a first-order ordinary differential equation exists uniquely is generalized to a system of equations.