Convert second-order ODE to first order system

Question

I want to convert the equation $y'' + y' + \sin y = 0$ into a system of first order ODEs.

I said:

$$u = y\enspace v = y'$$ $$u' = y'\enspace v' = y'' = -y' - \sin y$$ Hence we have the system: $$u' = v$$ $$v' = -v -\sin u$$

Is this correct? Or should $u = \sin y$ at the beginning?

Answer 1

Let $x_1 = y$, hence

$$\begin{align} x_1' &= y' = x_2 \\ x_2' &= y'' = -x_2 - \sin(x_1) \end{align}$$

Your answer is correct.

This system can be written as

$$X' = \begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = AX + f = \begin{pmatrix} 0 & 1 \\ 0 & - 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} 0 \\ -\sin(x_1) \end{pmatrix}$$