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Homework Help: Calculus 4. Pursuit Curve. Dog Chases Rabbit.

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data

    (a) In Example 1.18, 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?

    2. Relevant equations

    There are many integration techniques that could be used.

    3. The attempt at a solution

    Example 1.18:

    $y''+k^2y=0$ Where $k$ is an unknown real constant)

    We notice that the independent variable is missing. So we let:

    $$y'=p, y''=p\frac{dp}{dy}$$




    $$p=\pm k\sqrt{E-y^2}$$

    Now re-substitute $p=\frac{dy}{dx}$ to obtain:

    $$\frac{dy}{dx}= \pm k\sqrt{E-y^2}$$

    $$\frac{dy}{\sqrt{E-y^2}}= \pm kdx$$

    $$\sin^{-1}({\frac{y}{\sqrt{E}})}= \pm kx+F$$

    $$\frac{y}{\sqrt{E}}=\sin{( \pm kx+F)}$$

    $$y=\sqrt{E}\sin( \pm kx+F)$$

    Now we apply the sun formula for sine to rewrite the last expression as:

    $$y=\sqrt{E}\cos(F)\sin( \pm kx)+\sqrt{E}\sin(F)\cos( \pm kx)$$

    We may write a general solution as $y=A\sin(kx)+B\cos(kx)$

    There is another example in the book that I feel relates to this problem:

    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 $x$.




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



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



    Now, solve for $p$:


    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:


    Again, this is all I have to go on. I need to answer questions (a) and (b) stated at the top.
    Last edited: Sep 20, 2012
  2. jcsd
  3. Sep 20, 2012 #2
    could somebody at least give me something to go off of? This isn't making any sense.
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