Linear Systems of First-Order Differential Equations 379The eigenvalues are 1+2i = o+{3i, 1-2i, and 2 and the associated eigenvectors
are
= u+iv ,
Hence, the real general solution of (21) is
y(x) = ci[(ex cos 2x )u-(ex sin 2x)v] +c2[(ex sin 2x)u+(ex cos 2x)v] +c3e^2 xx3.
EXERCISES 8.3
In exercises 1-4 rewrite each of the linear systems of first-order
differential equations using matrix-vector notation.
1. y~ = 2y1 - 3y2 + 5ex 2. Y~ = Y2 - 2y1 + sin2x
Y~ = Y1 + 4y2 - 2e-x y~ = -3y1 + Y2 - 2 cos 3x
3. y~ = 2y2 4. y~ = 2xy1 - x^2 y2 + 4x
y~ = 3y1 y~ = exy1 + 3e-xy2 - cos 3x
Y~ = 2y3 - YI
- Consider the homogeneous linear system
(22) y / = (2
1-3) -2 y.
a. Verify thatY1 = G) ex and Y2 = G) e-x
are linearly independent solutions of (22).
b. Write the general solution of (22).
c. Verify thatis a particular solution of the nonhomogeneous linear system(23) YI = (2 1 -3) -2 y + (4X -3x 2) ·
d. Write the general solution of (23).