Expressing higher order ODE's as a system of First order ODE's

Lemma: An ${\text{n}}^{\text{th}}$ order differential equation written in explicit form can be written as a system of $\text{n}$ first order differential equations. Proof We have

$$y^{(n)}=f(t,y,y\prime ,y\prime \prime ...,y^{(n-1)})$$

Defining function $q$ such that

$$\begin{array}{rl}y& =q_0\\ q_0\prime & =q_1=y\prime \\ & .\\ & .\\ & .\\ q{\prime }_{(n-2)}& =q_{(n-1)}\\ q{\prime }_{(n-1)}& =q_n=f(t,q_0,q_1,...q_{n-1})\end{array}$$

This is useful because computers are good at solving first order systems of equations.¹

---

References

Footnotes

1. 4.1 Reducing a higher order DE to a system | DarrenOng1 ↩