SUGGESTED METHOD OF UNDETERMINED COEFFICIENTS
ANDREA BONITO
Given a, b, c ∈ R with a 6= 0, this is brief note discusses how to determine a
particular solution of
ay 00 + by 0 + cy = g
when g(t) = Pn (t) or g(t) = Pn (t)eβt or g(t) = Pn (t) cos(ωt) or g(t) = Pn (t) sin(ωt)
and Pn (t) = αn tn + αn−1 tn−1 + ... + a1 t + a0 is a polynomial of degree n ≥ 0.
1. Case: g(t) = Pn (t)
In order to match a polynomial of degree n on the right hand side, a particular
solution of
ay 00 + by 0 + cy = Pn
should read

 Qn (t)
Qn+1 (t)
yp (t) =

Qn+2 (t)
if c 6= 0,
if c = 0 and b 6= 0,
if c = 0 and b = 0.
Here Qn , Qn+1 and Qn+2 are polynomials of degree n, n + 1 and n + 2 respectively.
As noted in class (check it!), we can actually restrict the last two to be Qn+1 (t) =
tQn (t) and Qn+2 (t) = t2 Qn (t).
2. Case g(t) = Pn (t)eβt
To deal with this case, we advocate the change of variable
y(t) = v(t)eβt
so that the equation satisfied by v(t) is (check it!)
av 00 + p0 (β)v 0 + p(β)v = Pn ,
where p(t) = at2 + bt + c is the characteristic polynomial.
Applying the previous method we choose

if p(β) 6= 0,
 Qn (t)
tQn (t)
if p(β) = 0 and p0 (β) 6= 0,
vp (t) =
 2
t Qn (t)
if p(β) = 0 and p0 (β) = 0.
Once vp (t) is found, we deduce yp according to the formula
yp (t) = vp (t)eβt .
Date: March 23, 2015.
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ANDREA BONITO
3. Case g(t) = Pn (t) cos(ωt)
In order to find a particular solution of
ay 00 (t) + by 0 (t) + cy(t) = Pn (t) cos(ωt)
for ω =
6 0, we first note that
Pn (t) cos(ωt) = Re(Pn (t)eiωt )
and if z(t) = u(t) + iw(t) is a complex valued function satisfying
az 00 (t) + bz 0 (t) + cz(t) = Pn (t)eiωt ,
then y(t) = Re(z(t)) (check it!).
Therefore, we proceed as in the previous section and use the (complex valued)
function v(t) defined by
z(t) = v(t)eiωt ,
which satisfies
av 00 + p0 (iω)v 0 + p(iω)v = Pn .
Hence we can choose
Qn (t)
if p(iω) 6= 0,
vp (t) =
tQn (t)
otherwise,
where Qn (t) is a polynomial of degree n with complex coefficients. As a consequence
zp (t) = vp (t)eiωt
and
yp (t) = Re vp (t)eiωt .
4. Case g(t) = Pn (t) sin(ωt)
Proceed as in the previous section upon noting that
Pn (t) sin(ωt) = Im(Pn (t)eiωt )
so that
yp (t) = Im vp (t)eiωt .