second order ode?

λ² - 2λ + 1 = 0
λ = 1 , 1
yh = c1*e^t + c2*te^t

Let y1 = e^t , y2 = te^t , f = (e^t)/t

W( y1 , y2 )
= y1*y2' - y2*y1'
= e^t*( e^t + te^t ) - te^t*e^t
= e^(2t)

u1'
= - y2 * f / W
= - te^t * [ (e^t)/t ] / e^(2t)
= - 1

u1 = - t

u2'
= y1 * f / W
= e^t * [ (e^t)/t ] / e^(2t)
= 1/t

u2 = ln t

yp
= u1*y1 + u2*y2
= - t*e^t + ln t*te^t
= te^t( ln t - 1 )

y
= yh + yp
= c1*e^t + c2*te^t + te^t( ln t - 1 )

Ans: y = c1*e^t + c2*te^t + te^t( ln t - 1 )