Know Your Annihilators!
The inhomogeneous differential equation with constant coefficients
an y (n) + an−1 y (n−1) + · · · + a1 y 0 + a0 y = f (t)
d
.
dt
The following table lists all functions annihilated by differential operators with constant coefficients. The
table is also a table of general solutions to the homogeneous equation A(D)y = 0.
Remarks. (i) Throughout we let A and B be real constants and p(t) and q(t) denote polynomials of degree k.
(ii) Since any annihilator is a polynomial A(D), the characteristic equation A(r ) will in general have real roots
r and complex conjugate roots λ ± iω, possibly with multiplicity.
(iii) The differential operator whose characteristic equation has complex roots λ ± iω is then
can also be written compactly as P (D)y = f , where P (D) is a polynomial in D =
(D − (λ + iω))(D − (λ − iω)) = (D − λ)2 + ω2 = D 2 − 2λD + λ2 + ω2 .
Polynomials
1.
tk
D k+1
2.
p(t), q(t) polynomials of degree k
D k+1
Exponentials
3.
Aet
D−1
4.
Ae−t
D+1
5.
Aer t
D−r
6.
Aer t + Be−r t
D2 − r 2
7.
A cosh r t + B sinh r t
D2 − r 2
Trigonometric
8.
A cos t + B sin t
D2 + 1
9.
A cos ωt + B sin ωt
D 2 + ω2
10.
Aeλt cos ωt + Beλt sin ωt
(D − λ)2 + ω2
11.
p(t)er t
(D − r )k+1
12.
p(t)er t + q(t)e−r t
(D 2 − r 2 )k+1
13.
p(t) cosh r t + q(t) sinh r t
(D 2 − r 2 )k+1
14.
p(t) cos t + q(t) sin t
(D 2 + 1)k+1
15.
p(t) cos ωt + q(t) sin ωt
(D 2 + ω2 )k+1
16.
p(t)eλt cos ωt + q(t)eλt sin ωt
((D − λ)2 + ω2 )k+1
Multiplicity
Shift Principle
17.
P (D − r )[er t f (t)] = er t [P (D)f (t)]
18.
P (D)[er t f (t)] = er t [P (D + r )f (t)]
∆ Identities, ∆ = D 2 + ω2
19.
∆(f (t) cos ωt) = f 00(t) cos ωt − 2ωf 0(t) sin ωt
20.
∆(f (t) sin ωt) = f 00(t) sin ωt + 2ωf 0(t) cos ωt
Special Case of the ∆ Identities
If f (t) = at + b, then f 00 (t) = 0 and
21.
∆(f (t) cos ωt) = −2aω sin ωt
22.
∆(f (t) sin ωt) =
2aω cos ωt
Applications of the Shift Principle
Example 1. Apply the operator D + 3 to y = t 3 et .
(D + 3)[t 3 et ] = et [((D + 1) + 3)(t 3 )]
= et [(D + 4)(t 3 )]
= et (3t 2 + 4t 3 )
Example 2. Apply the operator P (D) = D 2 to y = t 3 et .
D 2 [t 3 et ] = et [(D + 1)2 (t 3 )]
= et [(D 2 + 2D + 1)(t 3 )]
= et (6t + 6t 2 + t 3 )
Example 3. Apply the operator P (D) = D 2 to y = e−t cos 2t.
D 2 [t 3 et cos 2t] = et [(D − 1)2 (cos 2t)]
= et [(D 2 − 2D + 1)(cos 2t)]
= et (−4 cos 2t + 4 sin 2t + cos 2t)
= −3et cos 2t + 4et sin 2t
Example 4 (Using ∆). Apply the operator P (D) = D 2 to y = e−t cos 2t. The annihilator of cos 2t is
∆ = D 2 + 4 so D 2 = ∆ − 4. Thus,
D 2 [et cos 2t] = et [(D − 1)2 (cos 2t)]
= et [(D 2 − 2D + 1)(cos 2t)]
= et [((∆ − 4) − 2D + 1)(cos 2t)]
= et [(∆ − 2D − 3)(cos 2t)]
= et (4 sin 2t − 3 cos 2t)
(since ∆(cos 2t) = 0)
= −3et cos 2t + 4et sin 2t
Example 5 (Using ∆). Apply the operator P (D) = D 2 to y = t 3 e−t cos 2t. Thus,
D 2 [t 3 et cos 2t] = et [(D − 1)2 (t 3 cos 2t)]
= et [(D 2 − 2D + 1)(t 3 cos 2t)]
= et [((∆ − 4) − 2D + 1)(t 3 cos 2t)]
= et [(∆ − 2D − 3)(t 3 cos 2t)]
= et [∆(t 3 cos 2t) − 2D(t 3 cos 2t) − 3t 3 cos 2t]
= et [(6t cos 2t − 4 · 3t 2 · sin 2t) − 2(3t 2 cos 2t − 2t 3 sin 2t) − 3t 3 cos 2t]
= et [6t cos 2t − 12t 2 sin 2t − 6t 2 cos 2t + 4t 3 sin 2t − 3t 3 cos 2t]
= (6t − 6t 2 − 3t 3 )et cos 2t + (−12t 2 + 4t 3 )et sin 2t