## 8.13. Special Methods For Determining A Particular Solution
Of The Nonhomogeneous Equation.
Reduction To A System Of First-Order Linear Equations

Although the method variation of parameters
can always provide a particular solution to the nonhomogeneous equation, it may
not be the simplest way to do so in specific cases.

### Example

Find a particular solution of

(
D−1
)(
D−2
)y=x
e
x+
x
2
(8.25)

#### Solution

Let
u=(
D−2
)y
. Then (8.25) becomes

(
D−1
)u=x
e
x+
x
2

which is a 1^{st} order equation
that can be solved using Theorem 8.1 or by inspection. This gives

u=
1
2
e
x+
x
2

so that

(
D−2
)y=u
=
1
2
e
x+
x
2

Using Theorem 8.1, we have, with
y
1
(
0
)=0
,

y
1
(
x
)=
1
2
e
2x
∫
0
x
dt
e
t
2
−t

Even though the integral cannot be
evaluated in terms of elementary functions, we can consider the problem
solved. The general solution is

y(
x
)=
c
1
e
x
+
c
2
e
2x
+
1
2
e
2x
∫
0
x
dt
e
t
2
−t