> [!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]]. # Properties **Methods of solution**: The [[Regular Perturbation Method for Ordinary Differential Equation|regular perturbation method]] 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. We find the functions $y_{0}, y_{1}, y_{2},\dots$ by substituting into the ODE. The dominant behaviour comes from the term, $y_{0}(t)$, the leading order term, solving the unperturbed problem. The regular perturbation method is considered successful if the difference between the approximate and exact solutions converges to zero at some defined rate as $\varepsilon \to 0$, uniformly on $(\alpha,\beta)$ with ($\epsilon<\epsilon_{0}$ for some $\epsilon_{0}$). **Singular perturbation methods** arise when the regular perturbation methods fail (Mahaffy, 2019).