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@jemoka / Jemoka Knowledge Base / raw/concept/kbhsecond_order_differential_equations.md
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--- title: "second order differential equation" source: https://www.jemoka.com/posts/kbhsecond_order_differential_equations/ --- the trick Here is a pretty ubiquitous trick to solve differential equations of the second order differential equations. It is used to change a second order differential equation to a First-Order Differential Equations. If you have a differential equation of the shape: \begin{equation} x^{’’} = f(x,x’) \end{equation} that, the second derivative is strictly a function between the first derivative value and the current value. We are going to define a notation \(x’ = v\), which makes sense. So, we will describe: \begin{equation} x^{’’} = \dv{v}{t} = \dv{v}{x} \dv{x}{t} = v\dv{v}{x} \end{equation} So therefore, we have: \begin{equation} x^{’’} = v\dv{v}{x} = f(x,v) \end{equation} So turns out, the original input \(t\) is, given a specific equation above, we have no need to know it. To actually go about solving it, see solving homogeneous higher-order differential equations.