Case Four: Repeated Complex Roots
Chapter One Hundred Twenty-Two
Syllabus topic Module 2, "2.3 Linear Differential Equations with Constant Coefficients"
Pages 271 to 272 of 303
In one line
A complex pair repeated k times gives the same cosine and sine terms multiplied by 1, x, and so on up to x to the power k minus 1.
The case
The two previous chapters combined. The auxiliary equation has a complex pair, and that pair is a repeated root.
That requires an equation of at least fourth order, because a repeated complex pair accounts for four roots. So this case is the one MU's "different cases" label reaches last, and it is set on fourth-order equations.
The answer
For the pair a plus or minus ib repeated twice:
y = e^(a x)((A + B x) cos(b x) + (C + D x) sin(b x))
Four arbitrary constants for a fourth-order equation, exactly as the order demands.
Why
Both previous reasons at once. The complex pair gives the cosine and sine, by Euler's formula. The repetition gives the extra factor of x, by the substitution argument of the repeated-roots chapter.
Formally: the operator contains ((D minus a) squared plus b squared) squared, and putting y = u e to the ax reduces the equation to (D squared plus b squared) squared u = 0, whose solutions are the cosine and sine of bx each multiplied by a linear function of x.
Worked
Solve y'''' plus 2y'' plus y = 0.
The auxiliary equation is m to the fourth plus 2m squared plus 1 = 0, which is (m squared plus 1) squared = 0.
So m squared = minus 1 twice, giving m = plus or minus i, each repeated. Here a is 0 and b is 1.
d4y/dx4 + 2 d2y/dx2 + y = 0
y = C1 cos(x) + C2 sin(x) + C3 x cos(x) + C4 x sin(x)
Four terms, four constants. The x cos x and x sin x terms grow, so this system's oscillation builds up without limit, which is exactly the resonance the Laplace chapter met from the other direction: driving an oscillator at its own frequency produces this equation.
Worked, with a real part
Solve the fourth-order equation whose auxiliary equation is ((m plus 1) squared plus 4) squared = 0.
Expanding, m squared plus 2m plus 5 repeated, so the roots are minus 1 plus or minus 2i, each twice. Here a is minus 1 and b is 2.
d4y/dx4 + 4 d3y/dx3 + 14 d2y/dx2 + 20 dy/dx + 25y = 0
y = C1 e^(-x) cos(2x) + C2 e^(-x) sin(2x) + C3 x e^(-x) cos(2x) + C4 x e^(-x) sin(2x)
The coefficients of that differential equation come from expanding (m squared plus 2m plus 5) squared, which is m to the fourth plus 4m cubed plus 14 m squared plus 20m plus 25.
Case Four: Repeated Complex Roots
(m^2 + 2m + 5)^2 = m^4 + 4 m^3 + 14 m^2 + 20 m + 25
Here the exponential decays and the x grows, and the exponential wins, so the oscillation rises and then dies. That is a critically damped oscillation.
All four cases, in one table
| Roots | Complementary function |
|---|---|
| m1, m2 real and distinct | A e^(m1 x) + B e^(m2 x) |
| m repeated twice | (A + Bx) e^(mx) |
| m repeated k times | (A + Bx + ... ) e^(mx), up to x^(k-1) |
| a plus or minus ib | e^(ax)(A cos bx + B sin bx) |
| a plus or minus ib, repeated twice | e^(ax)((A + Bx) cos bx + (C + Dx) sin bx) |
That table is the whole of MU's "different cases" label, and it is the single most useful page of section 2.3.
The rule that covers all of them
Every case is the same rule, which is worth stating once so that the five rows above stop being five things to remember.
Each root contributes e to the root times x. A root repeated k times contributes that exponential multiplied by 1, x, up to x to the power k minus 1. A complex pair's two exponentials are then combined into a cosine and a sine by Euler's formula.
That one sentence generates the table.
Check yourself
d4y/dx4 - y = 0
y = C1 e^x + C2 e^(-x) + C3 cos(x) + C4 sin(x)
d4y/dx4 + 8 d2y/dx2 + 16y = 0
y = C1 cos(2x) + C2 sin(2x) + C3 x cos(2x) + C4 x sin(2x)
The first has auxiliary equation m to the fourth minus 1 = 0, whose roots are 1, minus 1, i and minus i: two real and one complex pair, so two exponentials and a cosine and sine. The second is (m squared plus 4) squared = 0, a repeated complex pair.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.