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