00:02
All right, this would be linear if this was y prime and y, so i'm going to make a substitution.
00:10
I'm going to let u be y prime, and so u prime will be y double prime.
00:16
So this is xu prime minus u equals ln x over x, u prime minus 1 over x, u prime minus 1 over x, u equals lnx over x squared.
00:31
So now it's linear in u.
00:35
So we need an integrating factor.
00:38
That will be e to the minus 1 over x integral.
00:43
So e to the minus lnx, e to the lnx to the minus 1.
00:50
So x to the minus 1 over 1 or 1 over x.
00:56
So we have 1 over x u prime minus 1 over x squared u equals lnx over x over x squared u equals lnx over x cubed.
01:08
So this side should be 1 over xu prime, first derivative of the second plus second derivative of the first.
01:19
All right, that's all good.
01:21
So i'm going to integrate both sides here.
01:26
So on this side, i get 1 over xu.
01:28
And on this side, i'm going to have to do integration by parts.
01:34
So i'm going to think of this as x2 minus 3, lnx, dx...