Verifying a solution to DE. totally stuck

[darryw]

Homework Statement

verify that y_1(t) = t is solution of t^2y'' - t (t+2)y' + (t+2)y = 0

use "reduction of order to find 2nd linerary independent solution of equation

Homework Equations

I am stuck on the first part. Confused as to how (which method) I am expected to use to verify.
im stuck on getting the characteristic because of all the extra t's in the eqn.
im used to something with just y's like this: y'' + 5y' -7y = 0
but i don't know how to treat the t's

also when i try getting integrating factor, it doesn't work because its 2nd order (right?)
please help I am totally stuck thanks

The Attempt at a Solution

 

[gabbagabbahey]

Confused as to how (which method) I am expected to use to verify.

Simple. Substitute y=t into your differential equation...if you get 0=0 as a result, then it satisfies the DE.

im stuck on getting the characteristic because of all the extra t's in the eqn.

This isn't a constant coefficients problem, you aren't looking for the characteristic polynomial. You are supposed to use reduction of order instead.

Basically, once you verify that y_1(t)=t is a solution, you look for a second, linearly independent solution, of the form y_2(t)=u(t)y_1(t)=tu(t) by substituting this assumed form into your DE and solving the resulting DE you get for u(t).

I'm sure your textbook covers this method, so if you have difficulties, try opening your textbook and reading the relevant section

 

[darryw]

Thanks I figured this out. btw the textbook for this class is $197! its crazy. Professor always seem to make it so complicated too, so i just learn it all on youtube.
thanks for the help