Reduce Order (diff eq)

  • #1
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[SOLVED] Reduce Order (diff eq)

nevermind i got it, cliffsnotes ftw

Homework Statement


Solve the differential equation using the reduction of order method.
[tex]t^2 y'' - 4ty' + 6y = 0[/tex]
[tex]t > 0[/tex]
[tex]y_1 (t) = t^2[/tex]

Homework Equations





The Attempt at a Solution


Well The first thing I do is
[tex]y(t) = v(t) t^2[/tex]
Then I find y' and y'' and plug into the original diff eq and get
[tex]t^4 v'' = 0[/tex]
Which I'll assume is correct.

But now I'm really not sure what to do with that. I could do integration by parts? but that doesn't seem to lead anywhere. How do I get from there to t^3 (the other solution)?
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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I'm glad you solved it yourself- you would have felt embarassed if someone else had pointed out the obvious!

You divide both sides by t4 to get v"= 0. Since v" is 0, v'= C, a constant. Then v= Ct+ D, another constant. Since y= vt2, y= Ct3+ Dt2 is the general solution. By the way, we can divide by t4 only if t is not 0. The equation is singular at t= 0- the existance and uniqueness theorem does not apply if we are given intial conditions at t= 0. For example, there is no solution if we are given y(0)= 1, y'(0)= 0.
 

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