Reducing a Second-Order ODE

A second-order ODE of the form $y^{\prime \prime }=f(t,y,y^{\prime })$ can be reduced to a system of two first-order ODEs by introducing auxiliary variables.

Substitution

Set

$$z_1=y,z_2=y^{\prime }$$

Then

$$z_1^{\prime }=z_2,z_2^{\prime }=f(t,z_1,z_2)$$

This gives a two-component first-order system

$${\mathbf{z}}^{\prime }=\left(\begin{array}{c}{z}_2\\ f(t,z_1,z_2)\end{array}\right)$$

which can be solved with any standard method (e.g. Explicit Euler method, Fourth order Runge-Kutta).

General Case

An $n$th-order ODE $y^{(n)}=f(t,y,y^{\prime },\dots ,y^{(n-1)})$ becomes an $n$-component first-order system by setting $z_m=y^{(m-1)}$ for $m=1,\dots ,n$:

$$z_m^{\prime }=z_{m+1}(m<n),z_n^{\prime }=f(t,z_1,\dots ,z_n)$$

This reduction is essential for applying standard numerical ODE solvers to higher-order problems such as Newton’s second law.

Explicit Euler method | Fourth order Runge-Kutta | Runge-Kutta methods