66 Ordinary Differential Equations
Next, we consider the nonhomogeneous linear first-order differential equa-
tion
(28) y' = a(x)y + b(x)
where a( x) and b( x) are assumed to be defined and continuous on the interval
(a, {3) and b(x) -=/=. 0 on (a, {3). (Here "b(x) -=/=. 0 on (a , {3)" is read "b(x) is
not identically equal to zero on the interval (a, {3)" and equation (28) is call ed
nonhomogeneous because b(x) -=/=. 0 on (a, {3).) Let YK(x) denote the solution
of the homogeneous linear differential equation y' = a(x)y associated with the
nonhomogeneous linear differential equation (28). The exact expression for
YJdx ), which contains one arbitrary constant K, is given expli citly by equa-
tion (25). On the interval (a, {3), the function YK(x) satisfies y'g = a(x)YK·
Suppose that Yp ( x) is any particular solution of the nonhomogeneous differ-
ential equation y' = a( x )y + b( x) on the interval (a, {3). Since YP ( x) is a
particular solution, it contains no arbitrary constants and it satisfies y~(x) =
a(x)yp(x)+b(x) on the interval (a, {3). Assume that z(x) is any solution of the
nonhomogeneous differential equation (28) y' = a(x)y + b(x) which is defined
on the interval (a,{3). Since z(x) satisfies (28), z'(x) = a(x)z(x) + b(x) on
(a ,{3). Now consider the function w(x) = z(x) -yp(x). Differentiating, we
find for all x E (a, {3),
w'(x) = z'(x) -y~(x) = {a(x)z(x) + b(x)} - {a(x)yp(x) + b(x)}
= a(x)(z(x) - yp(x)) = a(x)w(x).
That is, w(x) is a solution of the associated homogeneous differential equation
(23) y' = a(a:)y on (a ,{3). Since equation (25) is the solution of (23), there is
a specific value of K such that for all x E (a, {3)
w(x) = z(x) -yp(x) = Kef a(x)dx.Hence, any solution z(x) of the nonhomogeneous linear differential equation
(28) y' = a(x)y + b(x) on the interval (a, {3) has the form
(29) z(x) = YK(x) + yp(x) = K ef a(x ) dx + yp(x)
where K is an arbitrary constant and Yp(x) is any particular solution of the
associated homogeneous differential equation (28).
It would be nice if we had an expression for the particular solution yp(x)
appearing in equation (29) or a procedure for computing yp(x) given specific
functions a(x) and b(x). Since the function a(x) appears explicitly in the first
term of (29), we anticipate that the function b(x) will appear explicitly in the
particular solution, the second term of (29). Let us assume that a(x) and b(x)