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.

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