> [!definition] > > $ > \frac{d^{n}y}{dx^{n}} = f\paren{t, y, \cdots, \frac{d^{n - 1}y}{dt^{n - 1}}} > $ > > A [[Differential Equation|differential equation]] is an **ordinary** differential equation if it only involves ordinary [[Derivative|derivatives]]. > [!definition] > > An ordinary differential equation is **autonomous** (without external force) if the function $f$ does not explicitly depend on the independent variable $t$. > [!theorem] > > Every $n$-th order ODE can be written as a system of $n$ first order ODEs by chaining derivatives. > [!definition] > > Let $(L)$ be an ODE, and $y: I \subseteq \real$ be a [[Continuity|continuous]] function of $t$, continuously [[Derivative|differentiable]] $n$ times. $y$ is a **solution** to $(L)$ if it satisfies the conditions of $L$. > > If $(L)$ imposes additional constraints, then those constraints must lie within $I$. > [!definition] > > Let $f(x, y)$ be a function. $f$ is homogenous if there exists $d$ such that $f(tx, ty) = t^df(x, y)$. Usually, $f$ is a polynomial with each term having the same total degree.