Confused on how my professor did this last step, DiffEQ, 2nd OrdER

[mr_coffee]

Hello everyone!
My professor was going over a problem real fast for the exam and now that i went over it again, I'm lost on how he did this last step. He is using a method called Abels Theorum. THe problem says:
Find a second solution of the given differential equation:
t^2y''+3ty' + y = 0;
y1(t) = 1/t;

U could use Eulers, but he doesn't want us to use that method, you could also use reduction of order but that would be more work and he doesn't want us to use that method either. He wants us to use Abels therom which seems to be the simplest out of all of them. Here is my work and I noted where I was confused at the very bottom of the page:


http://suprfile.com/src/1/2bgbtx/lastscan.jpg

Thanks if your confused on my work, let me know and i'll explain!

 

[assyrian_77]

Are you sure that the line shouldn't be $ty_2=ln|t|+C$

 

[mr_coffee]

oooo,
w00t
yeah ur right! the t magically dissapears then reappears!
thanks again assyrian!