Question

How to express a 3rd order differential equation into three first order differential equations?

Explain.

Answer 1

Let's 3rd order differential equation is $y'''(t) = f(t, y, y', y'')$. let introduce new vector-function $\overrightarrow{x}(t)$

$\overrightarrow{x} = \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} y(t) \\ y'(t) \\ y''(t) \end{bmatrix}$

then

$\frac{d \overrightarrow{x}}{d t} = \frac{d }{d t} \begin{bmatrix} y(t) \\ y'(t) \\ y''(t) \end{bmatrix} = \begin{bmatrix} x_2(t) \\ x_3(t) \\ f(t, x_1(t), x_2(t), x_3(t) \end{bmatrix}$