The Complementary Function, and What It Physically Means
Chapter One Hundred Twenty-Four
Syllabus topic Module 2, "2.3 Linear Differential Equations with Constant Coefficients"
Pages 276 to 277 of 303
In one line
The complementary function is what the system does with nothing driving it, and the four cases of the auxiliary equation are its four possible behaviours.
MU's label
"The complimentary Function", her spelling. The object is the general solution of f(D)y = 0.
The four cases, gathered
| Roots of the auxiliary equation | Contribution to the CF |
|---|---|
| a real root r | A e^(rx) |
| a real root r repeated k times | (A + Bx + ...) e^(rx), up to x^(k-1) |
| a complex pair a plus or minus ib | e^(ax)(A cos bx + B sin bx) |
| that pair repeated k times | the same, each constant replaced by a polynomial of degree k-1 |
Read off the roots, write one entry per root or pair, and add them. That is the whole procedure, and the four chapters before this one derived each row.
Worked, three cases in one equation
Find the CF of the equation whose auxiliary equation is:
(m - 1)(m + 2)^2 (m^2 + 9) = 0
Count the roots first: one from the first factor, two from the squared one, and two from the quadratic, which is five. So the differential equation is of fifth order and the CF will have five constants.
The roots are 1, minus 2 twice, and plus or minus 3i.
- The root 1 gives C1 e to the x.
- The root minus 2 repeated gives (C2 plus C3 x) e to the minus 2x.
- The pair plus or minus 3i gives C4 cos 3x plus C5 sin 3x.
d2y/dx2 + 4 dy/dx + 4y = 0
y = C1 e^(-2x) + C2 x e^(-2x)
The block above checks the middle piece on its own, which is the checkable part; the full fifth-order equation's coefficients would come from expanding that product.
(m - 1)(m + 2)^2 (m^2 + 9) = m^5 + 3 m^4 + 9 m^3 + 23 m^2 + 0 m - 36
Reading the coefficients off that expansion gives the differential equation, which is how a question in the other direction is set: "write down the equation whose CF is ..." wants exactly this multiplication.
What it means physically
This is the part worth carrying away, because it makes the four cases memorable.
The CF is the system's own behaviour. Stop driving it, give it a push, and what it does is the CF. It is the transient: in nearly every engineered system it dies away, leaving only the response to the input.
| Roots | Behaviour | Name |
|---|---|---|
| real, both negative | settles without oscillating | overdamped |
| real and repeated, negative | settles as fast as possible without overshoot | critically damped |
| complex with negative real part | oscillates, each swing smaller | underdamped |
| purely imaginary | oscillates for ever | undamped |
| any root with positive real part | grows without limit | unstable |
The Complementary Function, and What It Physically Means
The last row is the one an engineer cares about most. A system is stable exactly when every root of its auxiliary equation has a negative real part, which is a statement about where certain complex numbers sit relative to the imaginary axis, and it is why Module 1's first half is in this syllabus.
The connection to the Laplace transform
Worth drawing, because it is the same fact twice.
In Module 1, solving a differential equation by the transform produced a rational Y(s) whose denominator was exactly f(s), the auxiliary polynomial with s for m. Its roots were the poles, and the partial fractions over those roots produced exactly the terms of the CF.
So the CF's four cases and the partial-fraction chapters' four cases are the same four cases:
| Root of f | CF term | Partial fraction |
|---|---|---|
| real distinct | an exponential | a distinct linear factor |
| real repeated | x times an exponential | a repeated linear factor |
| complex pair | a sine and a cosine | an irreducible quadratic |
| repeated complex pair | x times a sine or cosine | a repeated irreducible quadratic |
Noticing that correspondence is the single best way to make both halves of this paper stick.
The constants, and how many
As many as the order. A second-order equation has two, a fifth-order one five. Each root contributes one, and a root repeated k times contributes k.
Counting them is a free check: an answer with the wrong number of constants is wrong, whatever else is right about it.
Check yourself
Write the CF for each auxiliary equation.
| Auxiliary equation | Complementary function |
|---|---|
| roots 2 and 5 | A e^(2x) + B e^(5x) |
| root 3 twice | (A + Bx) e^(3x) |
| roots plus and minus 4i | A cos 4x + B sin 4x |
| roots -1 plus and minus i | e^(-x)(A cos x + B sin x) |
| roots 0, 0, 1 | A + Bx + C e^x |
d3y/dx3 - d2y/dx2 = 0
y = C1 + C2 x + C3 e^x
The last block is the fifth row's equation, whose auxiliary equation is m cubed minus m squared = 0, that is m squared(m minus 1) = 0.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.