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48 CHAPTER 2 Linear Second-Order Equations

EXAMPLE 2.3
x
2x
e and e are solutions of y − 3y + 2y = 0. Therefore, every solution has the form

2x
x
y(x) = c 1 e + c 2 e .

This is the general solution of y − 3y + 2y = 0.

If we want to satisfy the initial conditions y(0) =−2, y (0) = 3, choose the constants c 1 and
c 2 so that
y(0) = c 1 + c 2 =−2
y (0) = c 1 + 2c 2 = 3.

Then c 1 =−7 and c 2 = 5, so the unique solution of the initial value problem is
y(x) =−7e + 5e .
2x
x
The Nonhomogeneous Case
We now want to know what the general solution of equation (2.1) looks like when f (x) is nonzero
at least for some x. In this case, the differential equation is nonhomogeneous.
The main difference between the homogeneous and nonhomogeneous cases is that, for the
nonhomogeneous equation, sums and constant multiples of solutions need not be solutions.

EXAMPLE 2.4
We can check by substitution that sin(2x) + 2x and cos(2x) + 2x are solutions of the non-
homogeneous equation y + 4y = 8x. However, if we substitute the sum of these solutions,

sin(2x) + cos(2x) + 4x, into the differential equation, we find that this sum is not a solution.
Furthermore, if we multiply one of these solutions by 2, taking, say, 2sin(2x) + 4x, we find that
this is not a solution either.

However, given any two solutions Y 1 and Y 2 of equation (2.1), we find that their difference
Y 1 − Y 2 is a solution, not of the nonhomogeneous equation, but of the associated homogeneous
equation (2.2). To see this, substitute Y 1 − Y 2 into equation (2.2):

(Y 1 − Y 2 ) + p(x)(Y 1 − Y 2 ) + q(x)(Y 1 − Y 2 )

=[Y + p(x)Y + q(x)Y 1 ]−[Y + p(x)Y + q(x)Y 2 ]
1 1 2 2
= f (x) − f (x) = 0.
But the general solution of the associated homogeneous equation (2.2) has the form c 1 y 1 +
c 2 y 2 , where y 1 and y 2 are linearly independent solutions of the homogeneous equation (2.2).
Since Y 1 − Y 2 is a solution of this homogeneous equation, then for some numbers c 1 and c 2 :
Y 1 − Y 2 = c 1 y 1 + c 2 y 2 ,
which means that

Y 1 = c 1 y 1 + c 2 y 2 + Y 2 .
But Y 1 and Y 2 are any solutions of equation (2.1). This means that, given any one solution
Y 2 of the nonhomogeneous equation, any other solution has the form c 1 y 1 + c 2 y 2 + Y 2 for some
constants c 1 and c 2 .
We will summarize this discussion as a general conclusion, in which we will use Y p (for
particular solution) instead of Y 2 of the discussion.

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October 14, 2010 14:12 THM/NEIL Page-48 27410_02_ch02_p43-76