Find the specific solution of the differential equation?

Question

t dy/dt + 4y= t ln t

satisfying y(1)=2.

Answer 1

Multiply through by (t^3)dt:
t^4 dy + 4yt^3 dt = t^4 ln t (dt)
But this is:
d(y*t^4) = t^4 (ln t) dt
Integrate left side from (1,2) to (t,y)
Integrate right side from 1 to t (using parts), and note that '|' means 'evaluated at':
y*t^4|(t,y) - y*t^4|(1,2) = ((t^5*(ln t))/5 - t^5/(5*5))|t - ((t^5*(ln t))/5 - t^5/(5*5))|1
or:
y*t^4 - 2 = t^5*(ln t)/5 - t^5/25 - 0 + 1/25
or:
yt^4 = (t^5/5) * (ln t - 1/5) + 51/25
or:
y = (t/5)*(ln t - 1/5) + 51/(25t^4)

Answer 2

yea, that is just gonna take too long and i have no paper laying around. i know how to do it, so if im free tomorrow i just may, although for a first order, it shouldnt be to hard