The differential equation is linear iff... thus every linear scalar ode has the form $ \sum_{i=0}^{n} a_{i}(x)y^{(i)} (x) = s(t)$with functions $s, a_{i}: (\alpha, \beta) \to \mathbb{R}, i=0,\dots,n$. # Properties **Method of complementary function and particular integral**: See [[Superposition Principle]]. Matrix approach: [[Reduction of Order of Scalar Ordinary Differential Equations]]. Nonlinear equations - [[Bernoulli Equation|Bernoulli equations]] are special because they are non-linear ODEs with known exact solutions. # Applications Examples - [[Implicit Solution to First Order Linear Ordinary Differential Equation Initial Value Problem]]; - [[Second Order Linear Scalar Ordinary Differential Equation]]; [[Solution to Inhomogeneous Second Order Linear Scalar Ordinary Differential Equation with Real Coefficients]]; - [[Linear ODE with Constant Coefficients]]; - See [[Solution to Homogenous Linear 2 x 2 System of First Order Ordinary Differential Equations with Real Coefficients]]. Pertubation: [[Perturbed Linear Ordinary Differential Equation]]. Generalisation: [[Linear Recurrence Relation with Constant Coefficients]].