# Bounded Second Order Differential Equations

• desbro05

#### desbro05

Hello all. I am having a very serious problem. The question states:

Find the value(s) of δ such that the solution of the initial-value problem

y'' − 4y = sin x;

where y(0) = δ and y'(0) = 0

is bounded.

I have no problem "solving" the equation and getting y in terms of x and δ, but what does bounded mean in this case, and what values satisfy this condition?

What did you get for a solution?
Bounded in this case means there exist a real number M such that
|y|<M for all x in (-∞,∞)
so pick the δ that keeps y from being big

The solution is -1/10, but I can't figure out how to go about finding this answer.

What was your solution (y in terms of x and δ) to
y'' − 4y = sin x;

where y(0) = δ and y'(0) = 0

What must δ be to assure y is never big?

The solution is -1/10, but I can't figure out how to go about finding this answer.
Well, there's your first problem! The solution to the equation, which is what lurflurf was asking, is not a number, it is a function of x. What did you get as the general solution to the differential equation? What did you get as the solution to this "initial value problem" (it will depend on $\delta$). Which of the functions in that solution will get "larger and larger" (for x getting larger both positive and negative?). You need to make the cofficients of those functions 0.