## 8.10. The Relation Between The Homogeneous And
Nonhomogeneous Equations

### Theorem 8.10.

Let
L:
C
(
n
)
(
J
)→C(
J
)
be a linear differential operator of order *n*.
Let
u
1
,⋯,
u
n
be *n*
independent
solution of the homogeneous equation
L(
y
)=0
. Let *y*_{1}
be a particular solution of the nonhomogeneous equation
L(
y
)=R
,
where
R∈C(
J
)
. Then the general solution for
L(
y
)=R
is

y=f(
x
)
=
y
1
(
x
)+
∑
k=1
n
c
k
u
k
(
x
)
(8.16)

where
c
k
are constants.

#### Proof

Let *f*
be any solution of
L(
y
)=R
. Since *L*
is linear, we have

L(
f−
y
1
)=L(
f
)−L(
y
1
)
=R−R=0

Hence,
(
f−
y
1
)∈N(
L
)
so that
f−
y
1
=
∑
k=1
n
c
k
u
k
. QED.

### Comment

Thus, Theorem 8.10 breaks down the problem of solving a
nonhomogeneous equation into 2 parts.
One is to find a particular solution, the other is to solve the
homogeneous equation. This has a simple
geometric analogy: to find all points on a plane, we find a particular point on
the plane, then add to it all points on the parallel plane that goes through
the origin.