Second Order Linear Differential Equations

So far we have only considered the nonhomogeneous differential equation (1) in cases
where the nonhomogeneous term f is a constant multiple of one of the functions e rx , cos βx,
sin βx, e αx cos βx, e αx sin βx, or is a sum of such functions. In general, the method of
undetermined coefficients can be applied in cases where
f(x) = p(x)e rx
f(x) = p(x) cos βx, or p(x) sin βx,
f(x) = p(x)e αx cos βx, or p(x)e αx sin βx
where p is a polynomial, or where f is a sum of such functions. This follows from the
fact that the expression y ′′ + ay ′ + by applied to
z = A0 + A1x + A2x 2 + ···+ Anx n e rx
will result in an expression of the form P (x)e rx where P is a polynomial of degree n
(or less); y ′′ + ay ′ + by applied to
z = A0 + A1x + A2x 2 + ···+ Anx n cos βx
will result in an expression of the form P (x) cos βx + Q(x) sin βx where P and Q are
a polynomials of degree n (or less); and so on.
The general version of the method of undetermined coefficients can be summarized as
follows:
(1) If f(x) =p(x)e rx where p is a polynomial of degree n, then
z(x) = A0 + A1x + A2x 2 + ···+ Anx n e rx .
(2) If f(x) =p1(x) cos βx + p2(x) sin βx where p1 and p2 are polynomials of degrees
k and m, respectively, then
z(x) =(A0 + A1x + ···+ Anx n ) cos βx +(B0 + B1x + ···+ Bnx n ) sin βx
where n = max {k, m}.
(3) If f(x) =p1(x)e αx cos βx + p2(x)e αx sin βx where p1 and p2 are polynomials of
degrees k and m, respectively, then
z(x) =(A0 + A1x + ···+ Anx n ) e αx cos βx +(B0 + B1x + ···+ Bnx n ) e αx sin βx
where n = max {k, m}.
Note: If any term in z satisfies the reduced equation y ′′ + ay ′ + by = 0, then use xz as
the trial solution; if any term in xz satisfies the reduced equation, then x 2 z will give a
particular solution.
Here are some examples.
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