Linear Second-Order Equations

Abstract

We have seen in 1.1.5 that the mathematical model for the mass–spring oscillator is a linear differential equation of second order with the constant coefficients $m,\,b$ and k. In this chapter we will discuss the methods of solving linear second order differential equations the form:

$$\displaystyle a(t)y^{\prime\prime}+b(t)y^{\prime}+c(t)y=f(t),$$

(3.1)

where $y=y(t)$ is the unknown (the dependent variable), $a(t),b(t)$ and $c(t)$ are continuous functions on some interval I with $a(t)\neq 0$ for t in I. The function $f(t)$ usually known as the input or the forcing term and the solution $y(t)$ is known as the output or response. As we will see later, the output $y(t)$ depends on the input $f(t)$. Hereafter, we use t as the independent variable, y the independent variable with $y^{\prime}=\frac{dy}{dt},\,y^{\prime\prime}=\frac{d^{2}y}{dt^{2}}$ and so on. Also, without confusion, we write sometimes the functions $y,y_{1},\dots$ as $y(t),y_{1}(t),\dots$

The solution of equation (3.1) depends on the solution of the equation (called homogeneous):

$$\displaystyle a(t)y^{\prime\prime}+b(t)y^{\prime}+c(t)y=0,$$

(3.2)

and on the form of the term $f(t)$ on the right-hand side of (3.1). We first discuss the case where the coefficients are constants and then we investigate the case of variable coefficients.