> [!Definition] Definition (Perturbed Linear Equation) > A perturbed linear ordinary differential equation is a [[Scalar Ordinary Differential Equation|scalar equation]] of the form $F(x,y,y',\dots,y^{(n)},\varepsilon) = 0, \quad x \in (\alpha,\beta),$ > with given function $f: (\alpha, \beta)\times \mathbb{R}^{k+1} \to \mathbb{R}$ and a given interval $(\alpha,\beta) \subset \mathbb{R},$ where the equation $F(x,y,y',\dots,y^{(n)},0) = 0$, called the unperturbed problem, is [[Linear Scalar Ordinary Differential Equation|linear]]. assumes a solution to the ODE has the form $y_{\epsilon}(t) = y_{0}(t) + \varepsilon y_{1}(t) + \varepsilon^{2} y_{2} (t) +\dots$called an $\varepsilon$-power series of the solution.