Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 16
Linear Equations With
Constant Coefficients
Definition 16.1. The general second order linear equation with constant
coefficients is
ay ′′ + by ′ + cy = f(t) (16.1)
where a, b, and c are constants, and a ≠ 0 (otherwise (16.1) reduces to
a linear first order equation, which we have already covered), and f(t)
depends only on t and not on y.
Definition 16.2. The Linear Differential Operator corresponding to
equation (16.1)
L = aD 2 + bD + c (16.2)
where
D = d dt and D2 = d2
dt 2 (16.3)
is the same operator we introduced in example (15.10). We can also write
L as
L = a d2
dt 2 + b d dt + c (16.4)
In terms of the operator L, equation (16.1) becomes
Ly = f(t) (16.5)
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