# Pursuit Curve. Dog Chases Rabbit. Calculus 4.

(a) In Example 1.21, assume that $a$ is less than $b$ (so that $k$ is less than $1$) and find $y$ as a function of $x$. How far does the rabbit run before the dog catches him?

(b) Assume now that $a=b$, and find $y$ as a function of $x$. How close does the dog come to the rabbit?

Example 1.21

A rabbit begins at the origin and runs up the $y-axis$ with speed $a$ feet per second. At the same time, a dog runs at speed $b$ from the point $(c,0)$ in pursuit of the rabbit. What is the path of the dog?

Solution: At time $t$, measured from the instant both the rabbit and the dog start, the rabbit will be at the point $R=(0,at)$ and the dog at $D=(x,y)$. We wish to solve for $y$ as a function of $x$.

$$\frac{dy}{dx}=\frac{y-at}{x}$$

$$xy'-y=-at$$

$$xy''=-a\frac{dt}{dx}$$

Since the $s$ is a arc length along the path of the dog, it follows that $\frac{ds}{dt}=b$. Hence,

$$\frac{dt}{dx}=\frac{dt}{ds}\frac{ds}{dx}=\frac{-1}{b}\sqrt{1+=(y')^2}$$

$$xy''=\frac{a}{b}\sqrt{1+(y')^2}$$

For convenience, we set $k=\frac{a}{b}$, $y'=p$, and $y''=\frac{dp}{dx}$

$$\frac{dp}{\sqrt{1+p^2}}=k\frac{dx}{x}$$

$$\ln({p+\sqrt{1+p^2}})=\ln(\frac{x}{c})^k$$

Now, solve for $p$:

$$\frac{dy}{dx}=p=\frac{1}{2}((\frac{x}{c})^k-(\frac{c}{x})^k)$$

In order to continue the analysis, we need to know something about the relative sizes of $a$ and $b$. Suppose, for example, that $a \lt$ $b$ (so $k\lt$ $1$), meaning that the dog will certainly catch the rabbit. Then we can integrate the last equation to obtain:

$$y(x)=\frac{1}{2}\{\frac{c}{k+1}(\frac{x}{c})^{k+1}-\frac{c}{1-k}(\frac{c}{x})^{k-1}\}+D$$

Again, this is all I have to go on. I need to answer questions (a) and (b) stated at the top.

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You can use $\TeX$ on this site by enclosing formulas in dollar signs; single dollar signs for inline formulas and double dollar signs for displayed equations. You can see the source code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands". Here's a basic tutorial and quick reference. There's an "edit" link under the question. –  joriki Sep 20 '12 at 4:35
Thank you. I wasn't sure how to do that. The link you provided greatly helped me. I fixed everything the best I could. Can you or someone else help me now? Thanks again. –  Pink Panda Sep 20 '12 at 15:53
I could really use a hint or something. –  Pink Panda Sep 20 '12 at 18:53
I don't understand -- (a) and (b) refer to Example 1.18 but talk about variables $a$ and $b$ that don't occur in Example 1.18 -- but they do occur in Example 1.21. Perhaps you could explain more about the context? –  joriki Sep 20 '12 at 21:59
By the way, you can get the appropriate size for a pair of parentheses (or other pairs of delimiters, like brackets, braces, absolute value bars) by preceding them with \left and \right, respectively. –  joriki Sep 20 '12 at 22:08