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
![|y(x)| \le |y(x_0)|e^{-a(x - x_0)} + \frac{k}{a}[1 - e^{-a(x - x_0)}]](https://s0.wp.com/latex.php?latex=%7Cy%28x%29%7C+%5Cle+%7Cy%28x_0%29%7Ce%5E%7B-a%28x+-+x_0%29%7D+%2B+%5Cfrac%7Bk%7D%7Ba%7D%5B1+-+e%5E%7B-a%28x+-+x_0%29%7D%5D&bg=ffffff&fg=000000&s=0 "|y(x)| \le |y(x_0)|e^{-a(x - x_0)} + \frac{k}{a}[1 - e^{-a(x - x_0)}]")
for .
(c) Let 
satisfy the same equation as (E) but with forcing function 
. That is
,
where is continuous on [
). Show that if
for ,
then
![|z(x) - y(x)| \le |z(x_0) - y(x_0)|e^{-a(x - x_0)} + \frac{K}{a}[1 - e^{-a(x - x_0)}]](https://s0.wp.com/latex.php?latex=%7Cz%28x%29+-+y%28x%29%7C+%5Cle+%7Cz%28x_0%29+-+y%28x_0%29%7Ce%5E%7B-a%28x+-+x_0%29%7D+%2B+%5Cfrac%7BK%7D%7Ba%7D%5B1+-+e%5E%7B-a%28x+-+x_0%29%7D%5D&bg=ffffff&fg=000000&s=0 "|z(x) - y(x)| \le |z(x_0) - y(x_0)|e^{-a(x - x_0)} + \frac{K}{a}[1 - e^{-a(x - x_0)}]")
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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