nth Order Linear Differential Equations

#OrdinaryDifferentialEquations

Subjects: Ordinary Differential Equations
Links: Existence and Uniqueness of Solutions to Systems of Differential Equations
An $n$ th order linear differential equation is an equation of the form

\sum_{k = 0}^{n} P_{k} (t) y^{(k)} = G (t)

We assume that $P_{0}, \dots, P_{n}$ and $G$ are continuous real valued functions on some open interval $(α, β)$ , and that $P_{n}$ is nowhere zero in this interval. Then dividing by $P_{n}$ we get that

L [y] = y^{(n)} + \sum_{k = 0}^{n - 1} p_{k} y^{(k)} = g (t)

$L$ is a linear differential operator of order $n$ . The theory is analogous to the second order linear differential operator. $L$ is of the form

L = D^{n} + \sum_{k = 0}^{n - 1} p_{k} D^{k}

Existence and Uniqueness

If there are functions $p_{0}, \dots, p_{n - 1}$ and $g$ are continuous on the open interval $I$ . Then there exists exctly one solution $y = ϕ (t)$ of the differential equation $L [y] = g$ , that satisfies the initial conditions

y (t_{0}) = y_{0}, y^{'} (t_{0}) = y_{0}^{'}, \dots y^{(n - 1)} (t_{0}) = y_{0}^{(n - 1)}

where $t_{0}$ is a point on the interval $I$ . This solution exists throughout $I$ .