Page 160 - Schaum's Outline of Differential Equations

CHAP. 16] e At 143

and from Eq. (16.5), r(k) = a^ + a 0. Thus, dr(k) I cfk = a^ The eigenvalues of At are A,j = X 2 = 3t, which is a single
eigenvalue of multiplicity two. It follows from Theorem 16.2 that

Solving these equations for a : and OQ, we find that

Substituting these values into (1) and simplifying, we have

At
16.5. Find e for A =

Here n = 3. From Eqs. (16.4) and (16.5) we have

2
and r (X) = a 2X + a{k + a Q. Thus,

Since the eigenvalues of At are A,j = A^ = ^3 = 3t, an eigenvalue of multiplicity three, it follows from Theorem 16.2
that

The solution to this set of equations is

Substituting these values into (1) and simplifying, we obtain