How to rewrite a 2nd order nonlinear ODE into an initial value problem of first order?

Question

I am trying to rewrite the initial value problem

$x''(t) = -sin(x(t))$ with initial values $x(0)= \pi/2 $ and $x'(0) = 0 $ into and initial value problem of first order ODE system.

Later I will have to implement some methods (Explicit/Implicit Euler) to solve this IVP so any help would be very much appreciated.

I do know how to rewrite a linear 2nd order ODE into a system of 1st order of the form $ x''(t)+g(t)*x'(t)+s(t)x(t)=g(t)$ but here I not know how to proceed because of the nonlinearity.

Answer 1

The technique works exactly the same as it does for a second-order linear ODE; define $y = x'(t)$, and then you have a first-order system in two variables: \begin{align*} x'(t) &= y(t) \\ y'(t) &= - sin(x(t)) \qquad \text{(since $y' = x''$)} \end{align*}

I believe that this technique can always be applied so long as the original ODE is quasilinear (i.e., the equation is linear in the highest derivatives of the unknowns.)