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58 CHAPTER 2 Linear Second-Order Equations

The general solution is
4 2x
x
−3x
y(x) = c 1 e + c 2 e + e .
5
EXAMPLE 2.13
Find the general solution of y − 5y + 6y =−3sin(2x).

3x
2x

The general solution of y − 5y + 6y = 0is c 1 e + c 2 e .

We need a particular solution Y p of the nonhomogeneous equation. Derivatives of sin(2x)
are constant multiples of sin(2x) or cos(2x). Derivatives of cos(2x) are also constant multiples of
sin(2x) or cos(2x). This suggests that we try a particular solution Y p (x)= A cos(2x)+ B sin(2x).
Notice that we include both sin(2x) and cos(2x) in this first attempt, even though f (x) just has
asin(2x) term. Compute

Y (x) =−2A sin(2x) + 2B cos(2x) and Y (x) =−4A cos(2x) − 4B sin(2x).
p p
Substitute these into the differential equation to get
− 4A cos(2x) − 4B sin(2x) − 5[−2A sin(2x) + 2B cos(2x)]
+ 6[A cos(2x) + B sin(2x)]=−3sin(2x).
Rearrange terms to write
[2B + 10A + 3]sin(2x) =[−2A + 10B]cos(2x).
But sin(2x) and cos(2x) are not constant multiples of each other unless these constants are zero.
Therefore,
2B + 10A + 3 = 0 =−2A + 10B.
Solve these to get A =−15/52 and B =−3/52. A particular solution is
15 3
Y p (x) =− cos(2x) − sin(2x).
52 52
The general solution is
15 3
3x 2x sin(2x).
y(x) = c 1 e + c 2 e − cos(2x) −
52 52
The method of undetermined coefficients has a trap built into it. Consider the
following.

EXAMPLE 2.14
x
Find a particular solution of y + 2y − 3y = 8e .

x
Reasoning as before, try Y p (x) = Ae . Substitute this into the differential equation to obtain
x
x
x
x
Ae + 2Ae − 3Ae = 8e .
But then 8e = 0, which is a contradiction.
x
x
The problem here is that e is also a solution of the associated homogeneous equation, so
x
x
the left side will vanish when Ae is substituted into y + 2y − 3y = 8e .

Whenever a term of a proposed Y p (x) is a solution of the associated homogeneous equa-
tion, multiply this proposed solution by x. If this results in another solution of the associated
homogeneous equation, multiply it by x again.
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October 14, 2010 14:12 THM/NEIL Page-58 27410_02_ch02_p43-76