TEST FUNCTIONS FOR A CUBIC

Let the cubic equation be

,4s3 + Bs2 + Cs + D = 0 (A > 0)

Then

F0 = A, F1 = B, F2 = BC – AD, F3 = D(BG – AD)

The necessary and sufficient conditions for all the test functions to be positive are that A, B, D, and (BG — AD) be positive. It follows that G also must be positive.

TEST FUNCTIONS FOR A QUARTIC

Let the quartic equation be

As4 + Bs3 + Cs2 + Ds + E = 0 {A > 0)

Then the test functions are F0 = A, F1 = B, F2 = BG — AD, F3 — F2D — B2E, F4 = FSBE. The necessary and sufficient conditions for these test functions to be positive are

A, B,D, E> 0

and D{BC – AD) – B2E > 0 (3.3,50)

It follows that C also must be positive. The quantity on the left-hand side of (3.3,50) is commonly known as Bouth’s discriminant.

TEST FUNCTIONS FOR A QUINTIC

Let the quintic equation be

As5 + Bs4 + Cs3 + Ds2 + Es + F = 0 (A > 0)

Then the test functions are F0 = A, Fx = B, F2 — BG — AD, F3 = F2D – B(BE – AF), Ft = F3(BE – AF) – F2F, F& = FtF2F. These test functions will all be positive provided that

A, B, D, F, F2, Ft > 0

It follows that C and E also are necessarily positive.

COMPLEX CHARACTERISTIC EQUATION

There may arise certain situations in which some of the coefficients of the differential equations of the system are complex instead of real, and conse­quently some of the coefficients of the characteristic equation are complex too. The criteria for stability in that case are discussed by Morris (ref. 3.7).

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