yyttetSolutiony yt tet y yt tet Integrating factor e 1t

y\'-(y/t)=te^-t

Solution

y\' = y/t + t*e^(-t)

y\' - y/t = t*e^(-t)

Integrating factor = e^(- (1/t)dt) = e^(-lnt)=1/t

Multiply both sides of the D.E by 1/t.

(1/t)y\' - y/t^2 = e^(-t)

d/dt[y*(1/t)] = e^(-t)

y*(1/t) = e^(-t) dt

y=t*e^(-t) dt

use integration by parts again
u = t
dv = e^(-t) dt
du = dt
v = -e^(-t)

= -te^(-t) + e^(-t) dt
= -te^(-t) - e^(-t) +c

y= -te^(-t) - e^(-t)