# Reduction of Order Problem

#### praecox

Hey, guys. I'm having a hard time with a 2nd ODE reduction of order problem.

The DE is
t2y'' - 4ty' + 6y = 0, with y1 = t2.

So I set up my y = t2v(t); y' = t2v' + 2tv; y'' = t2v'' + 4tv' + 2v.
I then substituted back in and got:
t4v'' = 0

This is where I'm getting stuck. The book says that the answer I'm looking for is y2 = t3, but I can't seem to get there from here.

I don't think I can use an integrating factor here (if I'm wrong there let me know), so I tried separating the variables, but that got me no where. I think I did it wrong though.

Here was my separation of variables attempt:
letting w = v', so w' = v'':
t4dw/dt = 0
dw = t-4 dt
Integrating both sides (and taking constants of integration to be zero):
w = -1/3 t-3
since w = v':
dv = -1/3 t-3 dt
Integrating again to find v and still no where near t3.

Any ideas? Anybody? :/

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#### Some Pig

Let y=tn
n(n-1)tn-4ntn+6tn=0
(n-2)(n-3)tn=0
n=2 or 3

#### matematikawan

Hey, guys. I'm having a hard time with a 2nd ODE reduction of order problem.

The DE is
t2y'' - 4ty' + 6y = 0, with y1 = t2.

So I set up my y = t2v(t); y' = t2v' + 2tv; y'' = t2v'' + 4tv' + 2v.
I then substituted back in and got:
t4v'' = 0

Any ideas? Anybody? :/
v"(t)=0 hence v(t) is linear. Choose v(t)=t so that y = t2v(t)=t3.

#### praecox

That makes so much sense now! Thank you so much! :D