Behavior of y = -cos(t)/t² + (aπ²)/(4t²) as t → 0

[TheCarl]

Homework Statement

http://edugen.wileyplus.com/edugen/shared/assignment/test/session.quest1886032entrance1_N10020.mml?size=14&rnd=1360201586591

(b) Solve the initial value problem and find the critical value a₀ exactly.

y = ?​

a₀ = ?​

(c) Describe the behavior of the solution corresponding to the initial value a₀.

y -> ? as t -> 0​

The Attempt at a Solution

I got part b correct but I thought I'd put it in here to help speed the process for whoever can help me.

(b) y= -cos(t)/(t^2) + (a*pi^2)/(4t^2)

a₀ = 4/pi^2

(c) I would think y would approach 0 as t approaches 0 but that apparently is wrong. This is where I need assistance. Any help is greatly appreciated.

 

[haruspex]

(b) y= -cos(t)/(t^2) + (a*pi^2)/(4t^2)
a₀ = 4/pi^2

I assume a = a₀, so y= (1-cos(t))t^-2
Do you know an expansion for cos(t) valid in the vicinity of 0?

 

[TheCarl]

I apologize if I seem a bit dense but could you elaborate on your question about the expansion on cos(t)? I'm not entirely sure what you're asking.

 

[haruspex]

Taylor expansion? Power series?