# Quod Erat Demonstrandum

## 2009/02/17

### Asymptotic Behavior of Solutions to Linear Equations

Apart from the interesting article in the last post, Justin also sent me the following interesting question:

Consider

$\frac{dy}{dx} + ay = Q(x)$ – – – – – – (E)

where the constant $a$ is positive and $Q(x)$ is continuous on [$0,\infty$).

(a) Show that the general solution to equation (E) can be written in the form

$y(x) = y(x_0)e^{-a(x - x_0)} + e^{-ax}\int_{x_0}^xe^{at}Q(t)dt$

where $x_0$ is a non-negative constant.

(b) If $|Q(x)| \le k$ for $x \ge x_0$, where $k$ and $x_0$ are non-negative constants, show that

$|y(x)| \le |y(x_0)|e^{-a(x - x_0)} + \frac{k}{a}[1 - e^{-a(x - x_0)}]$ for $x \ge x_0$.

(c) Let $z(x)$ satisfy the same equation as (E) but with forcing function $\Tilde{Q}(x)$. That is

$\frac{dz}{dx} + az = \Tilde{Q}(x)$,

where $\Tilde{Q}(x)$ is continuous on [$0, \infty$). Show that if

$|\Tilde{Q}(x) - Q(x)| \le K$ for $x \ge x_0$,

then

$|z(x) - y(x)| \le |z(x_0) - y(x_0)|e^{-a(x - x_0)} + \frac{K}{a}[1 - e^{-a(x - x_0)}]$ for $x \ge x_0$.

(d) Show that if $Q(x) \rightarrow \beta$ as $x \rightarrow \infty$, then any solution $y(x)$ of (E) satisfies $y \rightarrow \frac{\beta}{a}$ as $x \rightarrow \infty$.
[Hint: Take $\Tilde{Q}(x) = \beta$ and $z(x) = \frac{\beta}{a}$ in part (c).]

(e) Suppose brine solution containing $q(t)$ kg of salt per litre at time $t$ 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 $q(t) \rightarrow \beta$ as $t \rightarrow \infty$, by using the result in part (d), determine the limiting concentration of the salt in the tank as $t \rightarrow \infty$.

I think the question is well set and suitable for self-study at secondary school standard, try.