0.2 Nonhomogeneous Linear Equations 15
that the name of the integration variable is changed fromtto something else
(here,z) to avoid confusing the limit with the dummy variable of integration.
The simple second-order equation
d^2 u
dt^2=f(t) (4)can be solved by two successive integrations.
The two theorems that follow summarize some properties of linear equa-
tions that are useful in constructing solutions.
Theorem 1.The general solution of a nonhomogeneous linear equation has the
form u(t)=up(t)+uc(t),whereup(t)is any particular solution of the nonho-
mogeneous equation and uc(t)is the general solution of the corresponding homo-
geneous equation.
Theorem 2.If up 1 (t)and up 2 (t)are particular solutions of a differential equation
with inhomogeneities f 1 (t)and f 2 (t),respectively,thenk 1 up 1 (t)+k 2 up 2 is a par-
ticular solution of the differential equation with inhomogeneity k 1 f 1 (t)+k 2 f 2 (t)
(k 1 ,k 2 are constants).
Example.
Find the solution of the differential equation
d^2 u
dt^2+u= 1 −e−t.The corresponding homogeneous equation is
d^2 u
dt^2 +u=^0 ,with general solutionuc(t)=c 1 cos(t)+c 2 sin(t)(found in Section 1). A par-
ticular solution of the equation with the inhomogeneityf 1 (t)=1, that is, of
the equation
d^2 u
dt^2 +u=^1 ,
isup 1 (t)=1. A particular solution of the equation
d^2 u
dt^2+u=e−tisup 2 (t)=^12 e−t. (Later in this section, we will review methods for constructing
these particular solutions.) Then, by Theorem 2, a particular solution of the
given nonhomogeneous Eq. (5) isup(t)= 1 −^12 e−t. Finally, by Theorem 1, the