# Systems of second order ODE's

by peterbone
Tags: order, systems
 Sci Advisor HW Helper PF Gold P: 12,016 Systems of second order ODE's Suppose you've got a second order diff.eq system: $$\frac{d^{2}\vec{Y}}{dt^{2}}=\vec{F}(y_{1},..y_{n},\dot{y}_{1},....,\dot {y}_{n},t), \vec{Y}(t)=(y_{1}(t),....,y_{n}(t)),\dot{y}_{m}\equiv\frac{dy_{m}}{dt}, 1\leq{m}\leq{n}; m,n\in\mathbb{N}$$ Now, define: $$\vec{X}(t)=(x_{1}(t),....,x_{n}(t),....,x_{2n}(t))$$ with: $$x_{i}=y_{i}, x_{n+i}=\frac{dy_{i}}{dt}=\frac{dx_{i}}{dt}, 1\leq{i}\leq{n}$$ Thus, we may form the 1-order differential system of 2n equations: $$\frac{d\vec{X}}{dt}=\vec{G}(\vec{X},t)$$ where: $$G_{i}(\vec{X},t)=x_{n+i}, 1\leq{i}\leq{n}$$ [tex]G_{i}(\vec{X},t)=F_{i-n}(\vec{X},t), n