Page 316 - A Course in Linear Algebra with Applications

3 0 0 Chapter Eight: Eigenvectors and Eigenvalues

X
equation Y' = AY becomes U' = {S~ AS)U = DU, which is
equivalent to

u
u[ = wi, ^2 — ~ 2, u' 3 = 2v,3, u' A = —2M 4.

Solving these simple equations, we obtain

x
x
2:r
2x
ui=cie , u 2 = c 2e~ , u 3 = c 3 e , •u 4 = c 4 e~ .
The functions 2/1 and 2/2 ma Y n o w De read off from the equation
Y = SU to give the general solution

2/1 = cie* + 2c 2e~ x + c 3e 2a; + c 4 e -2:E
= 2cie a: — C2e _x + c^e 2x — c$e~ 2x
y 2

Exercises 8.3

1. Find the general solutions of the following systems of linear
differential equations:

y
= Vl+
2 2
(a) {ti - * (b) ) = l yi 7 /
l
{V2= 2yi~ 3y 2 x {y 2 = - 2yi + 3y 2
2/1 = 2/1+2/2 + 2/3
(c) { y' 2= 2/2
y's = 2/2 + ys

2. Find the general solution (in real terms) of the system of
differential equations

y'i= 2/i+ 2/2
= —22/1 +
V 2 3y 2
Then find a solution satisfying the initial conditions yi(0) = 1,
2/ 2(0) = 2.