- #1
Ressurection
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Homework Statement
Given the differential equation : [itex]\frac{dy}{dt}[/itex] + a(t)*y = f(t)
in which a and f are continuous functions in ℝ and verify:
a(t) > c > 0 [itex]\forall[/itex]t , lim f(t) = 0
Show that any solution of the differential equation satisfies:
lim y(t) = 0
Homework Equations
The Attempt at a Solution
My first thought was to apply the limit to the equation right away, so that would give me
lim (y') + lim(a) * lim(y) = 0
Now this means both the limits of y and y' must exist, and since a(t) > c > 0, either they are both 0 or one is positive and the other negative.
When lim y is a finite constant, then lim y' = 0, which only allows for the case of both limits being 0. (Side note: does lim y being finite imply that lim y' = 0?)
When limit y is +∞, this would require limit y' = -∞. Is this case possible?
My intuition tells me it isn't, but I'm not absolutely sure..
Also, if lim (a) = ∞, then I don't see any restriction that imposes lim y = 0, which makes me believe there may be a more simpler way to solve this problem.
Note: All limits refer to t → ∞.