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We now return to the nonhomogeneous equation
L [y] = / + p(t) Ó + q (t) Ó = g(t), (1)
where p, q, and g are given (continuous) functions on the open interval I. The equation
L [y] = / + p(t) Ó + q (t) ó = 0, (2)
in which g(t) = 0 and p and q are the same as in Eq. (1), is called the homogeneous equation corresponding to Eq. (1). The following two results describe the structure of solutions of the nonhomogeneous equation (1) and provide a basis for constructing its general solution.
Chapter 3. Second Order Linear Equations
If Yt and Y2 are two solutions of the nonhomogeneous equation (1), then their difference Y( - Y2 is a solution of the corresponding homogeneous equation (2). If, in addition, y( and y2 are a fundamental set of solutions of Eq. (2), then
Yt(t) - Y2(t) = c(y((t) + c2y2(t), (3)
where c1 and c2 are certain constants.
To prove this result, note that Y1 and Y2 satisfy the equations
L [Y(](t) = g(t), L [Y2](t) = g(t). (4)
Subtracting the second of these equations from the first, we have
L[Y(](t) - L[Y2](0 = g(t) - g(t) = 0. (5)
L[Y(] - L[Y2] = L[Y( - Y2],
so Eq. (5) becomes
L[Y( - Y2](t) = 0. (6)
Equation (6) states that Y( - Y2 is a solution of Eq. (2). Finally, since all solutions of Eq. (2) can be expressed as linear combinations of a fundamental set of solutions by Theorem 3.2.4, it follows that the solution Yt - Y2 can be so written. Hence Eq. (3) holds and the proof is complete.
The general solution of the nonhomogeneous equation (1) can be written in the form
Ó = Ô(t) = c( y( (t) + c2y2(t) + Y (t), (7)
where yt and y2 are a fundamental set of solutions of the corresponding homogeneous equation (2), c( and c2 are arbitrary constants, and Y is some specific solution of the nonhomogeneous equation (1).
The proof of Theorem 3.6.2 follows quickly from the preceding theorem. Note that Eq. (3) holds if we identify Y( with an arbitrary solution ô of Eq. (1) and Y2 with the specific solution Y. From Eq. (3) we thereby obtain
ô() - Y(t) = c(y((t) + c2y2(t), (8)
which is equivalent to Eq. (7). Since ô is an arbitrary solution ofEq. (1), the expression
on the right side ofEq. (7) includes all solutions ofEq. (1); thus it is natural to call it
the general solution ofEq. (1).
In somewhat different words, Theorem 3.6.2 states that to solve the nonhomogeneous equation (1), we must do three things:
1. Find the general solution ctyt(t) + c2y2(t) of the corresponding homogeneous equation. This solution is frequently called the complementary solution and may be denoted by yc (t).
2. Find some single solution Y (t) of the nonhomogeneous equation. Often this solution is referred to as a particular solution.
3. Add together the functions found in the two preceding steps.
3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients
We have already discussed how to find yc ( t), at least when the homogeneous equation (2) has constant coefficients. Therefore, in the remainder of this section and in the next, we will focus on finding a particular solution Y(t) of the nonhomogeneous equation (1). There are two methods that we wish to discuss. They are known as the method of undetermined coefficients and the method of variation of parameters, respectively. Each has some advantages and some possible shortcomings.
Method of Undetermined Coefficients. The method of undetermined coefficients requires that we make an initial assumption about the form of the particular solution Y (t), but with the coefficients left unspecified. We then substitute the assumed expression into Eq. (1) and attempt to determine the coefficients so as to satisfy that equation. If we are successful, then we have found a solution of the differential equation (1) and can use it for the particular solution Y (t). If we cannot determine the coefficients, then this means that there is no solution of the form that we assumed. In this case we may modify the initial assumption and try again.
The main advantage of the method of undetermined coefficients is that it is straightforward to execute once the assumption is made as to the form of Y(t). Its major limitation is that it is useful primarily for equations for which we can easily write down the correct form of the particular solution in advance. For this reason, this method is usually used only for problems in which the homogeneous equation has constant coefficients and the nonhomogeneous term is restricted to a relatively small class of functions. In particular, we consider only nonhomogeneous terms that consist of polynomials, exponential functions, sines, and cosines. Despite this limitation, the method of undetermined coefficients is useful for solving many problems that have important applications. However, the algebraic details may become tedious and a computer algebra system can be very helpful in practical applications. We will illustrate the method of undetermined coefficients by several simple examples and then summarize some rules for using it.