Page 159 - Schaum's Outline of Differential Equations

142 e At [CHAP. 16

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16.2. Find e for A =
Since n = 2, it follows from Eqs. (16.3) and (16.5) that

and r (X) = c^X + a 0. The eigenvalues of At are Xj = 2t and X 2 = -4f, which are both of multiplicity one. Substituting
these values successively into (16.6), we obtain

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

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

At
16.3. Find e for A =

Here n = 2; hence,

and r(k) = cqX + a 0. The eigenvalues of At are Xj = it and X 2 = —it, which are both of multiplicity one. Substituting
these values successively into Eq. (16.6), we obtain

Solving these equations for a : and a 0 and using Euler's relations, we find that

Substituting these values into (_/), we obtain

16.4. Find e^' for A =

Here n = 2. From Eq. (16.3),