Apart from the interesting article in the last post, Justin also sent me the following interesting question:
Consider
– – – – – – (E)
where the constant is positive and
is continuous on [
).
(a) Show that the general solution to equation (E) can be written in the form
where is a non-negative constant.
(b) If for
, where
and
are non-negative constants, show that
for
.
(c) Let satisfy the same equation as (E) but with forcing function
. That is
,
where is continuous on [
). Show that if
for
,
then
for
.
(d) Show that if as
, then any solution
of (E) satisfies
as
.
[Hint: Take and
in part (c).]
(e) Suppose brine solution containing kg of salt per litre at time
runs into a tank of water at a fixed rate and that the mixture, kept uniform by stirring, flows out at the same rate. Given that
as
, by using the result in part (d), determine the limiting concentration of the salt in the tank as
.
I think the question is well set and suitable for self-study at secondary school standard, try.
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