Ordinary differential equations Proposition

Reduction of a higher-order equation to a first-order system

Builds on Ordinary differential equation Solution of an ordinary differential equation Initial value problem

Statement

Let $n \ge 1$ , let $U$ be a subset of $\mathbb{R} \times \mathbb{R}^n$ , and let $f : U \to \mathbb{R}$ be a function. Consider the equation $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$ and the first-order system for $z = (z_1, \dots, z_n)$ : $z_1' = z_2, \quad z_2' = z_3, \quad \dots, \quad z_{n-1}' = z_n, \quad z_n' = f(x, z_1, \dots, z_n).$ Let $I$ be an open interval. If $y$ is a solution of the equation on $I$ , then $z = \left(y, y', \dots, y^{(n-1)}\right)$ is a solution of the system on $I$ . Conversely, let $z$ be a solution of the system on $I$ . Then $y = z_1$ is a solution of the equation on $I$ , and $z_k = y^{(k-1)}$ for $k = 1, \dots, n$ .

Remarks

The initial conditions $y^{(k)}(x_0) = c_k$ for $k = 0, \dots, n-1$ become the one condition $z(x_0) = (c_0, \dots, c_{n-1})$ . So results on first-order initial value problems apply to equations of each order. For $y'' = -y$ , the system is $z_1' = z_2$ , $z_2' = -z_1$ . The same method changes a system of equations of order $n$ into a larger first-order system.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Initial value problem definition
Reduction of a higher-order equation to a first-order system statement

Tag	`0TH9CPW3VS`
URI	`m2://0TH9CPW3VS`
Kind	statement, proposition
Writing	W0 stub
MSC	34A99
Status	This copy of the site has no build data. Run `m2 status 0TH9CPW3VS`.
Uses	Ordinary differential equation, Solution of an ordinary differential equation, Initial value problem

Reduction of a higher-order equation to a first-order system

Statement

Remarks

Reading path

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