> [!definition] > > An [[Ordinary Differential Equation|ordinary differential equation]] is **linear** if > $ > \sum_{k = 0}^{n}a_k(t)\frac{d^k}{dt^k} - g(t) = 0 > $ > with each $a_k(t)$ and $g(t)$ being known [[Function|functions]] of the independent [[Variable|variable]]. In other words, it is a [[Linear Transformation|linear transformation]] on the space of derivatives of the dependent variable. > > Its solutions are a [[Vector Space|vector space]] of dimension $n$, which allows writing them as a [[Linear Combination|linear combination]] of [[Basis|basis]] (fundamental set of solutions). > [!definition] > > The linear ODE $(L)$ is **homogenous** if $g(t) = 0$. > [!definition] > > The linear ODE $(L)$ is **constant coefficient** if each $a_k(t)$ is constant.