A Critical Value for Transition in Differential Equations

[ehabmozart]

Homework Statement

Q) ty' + 2y = (sin t)/t, y(−π/2) = a, t < 0 . Let a₀ be
the value of a for which the transition from one type of behavior to another occurs.Solve the IVP and find the critical value a₀ exactly.

Homework Equations

DE

The Attempt at a Solution

I can easily manage to get the general solution. After a series of work, I ended with y= ∏²a/4t² - cost/t² . My biggest problem here is getting a₀, the critical value. I mean what is exactly the critical value. In this case I cannot equate y' to be zero. I want a general explanation on what the critical value means!?

 

[szynkasz]

Consider $\lim\limits_{t\to 0}y(t)$ in dependence on $a$