Page 161 - Schaum's Outline of Differential Equations

144 e At [CHAP. 16

At
16.6. Find e for A =

Here n = 3. From Eq. (16.4),

2
and from Eq. (16.5), r(k) = c^A, + c^A + a Q. The eigenvalues of At are A : = 0 and A 2 = Ag = f; hence A = f is an
eigenvalue of multiplicity two, while X = 0 is an eigenvalue of multiplicity one. It follows from Theorem 16.2 that
e' = r(t), e' = r'(t), and e° = r(0). Since r'(A) = la^k + a 1; these equations become

which have as their solution

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

16.7. Find e^' for A =

Here n = 3. From Eq. (16.4),

2
and from Eq. (16.5), r(k) = a 2X + a :X + a 0. The eigenvalues of At are A,j = 0, A^ = ;Y, and Xg = —rt. Substituting
these values successively into (16.6), we obtain the three equations