Linear Systems of First-Order Differential Equations 369are linearly independent solutions of the homogeneous linear system(8)on the interval (0, oo) and write the general solution of (8) on (0, oo).
SOLUTIONNotice that A(x) is not defined for x = 0, but A(x) is defined and con-
tinuous on (-oo, 0) and (0, oo ). So by the existence theorem, there are two
linearly independent solutions of (8) on (0, oo).
Differentiating y 1 , we findy~(x) = (~).
Multiplying A(x) by Y1(x), we get for x-/= 0
A (x)y,(x) ~ (: :!) C~) (
~(1)^2 + ~(2x) -1 )
2(1) + 0(2x)Since y~ = A(x)y 1 for x-/= 0, the vector y 1 is a solution of (8) on (0, oo).
Differentiating Y2 ( x), we findComputing A (x)y2(x), we obtain for x-/= 0
A(x)y2(x) = x x2 - x x2
(~ -l)(x ) (~(x)+-l(x2))
2 0 x
2- 2(x) + O(x^2 )