Verify that y_1(t) and y_2(t) are solutions of y'' - 2y' + y = 0 where y_1(t) = e^t and y_2(t) = te^t. Do they constitute a fundamental set?
Solution:
Transforming into characteristic equation the equation becomes:
r^2 - 2r + 1 = 0
(r - 1)^2 = 0
r = 1
Hence y(t) = c_1*e^t + c_2*te^t
w = e^t * (e^t + te^t) - (e^t * te^t)
= (e^t)^2 + e^t(te^t) - (e^t)(te^t)
= (e^t)^2
≠ 0
Hence they do constitute a fundamental set.
