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Brachistochrone homework problem

  1. Dec 1, 2008 #1
    1. use the parametric equations of a cycloid ( x=a(t-sint), and y=a(1-cost) ) to show that y=y(x) is the solution of the differential equation for any parameter a. Find the relationship between the radius a in the parametric equations and the constant C in y(1+y`2)=C.

    2. Solve the equation y(1+y`2)=C with the initial condition y(0)=0. Express rather x as the function of y. what is the interpretation of the constant C in terms of a cycloid.



    I need help starting the first question. In #2, im stuck at 1+y`2= C/y. i know your not supposed to subtract 1 to either side, so how am i supposed to isolate y` by itself?
     
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  3. Dec 1, 2008 #2

    Dick

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    Re: Brachistochrone

    To start the first part just substitute the forms you are given for x and y into the equation. y'=dy/dx=(dy/dt)/(dx/dt).
     
  4. Dec 1, 2008 #3
    Re: Brachistochrone

    if i sub those in, i get 1-cost=1-cos(t-sint). Im stuck here :(

    As for questions number 2, i got to x=[tex]\int[/tex][tex]\frac{2cu^{2}}{(1+u^{2})^{2}}du[/tex]. where do i go from here?
    (with u^2 = y/(c-y) )
     
  5. Dec 2, 2008 #4

    Dick

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    Re: Brachistochrone

    For part 1), no, you don't get that. Show your work. And I'm not dealing with the second part until you get the first.
     
  6. Dec 2, 2008 #5
    Re: Brachistochrone

    ok, im a bit confused here. Do u mean sub those parametric equations into y=y(x) or y(1+y`2)=C?

    if its y=y(x), thats how i got 1-cost=1-cost(t-sint).
    If its the latter, then dy/dx=sint/(1-cost).
    then i plug it into the equation to get 1-cost(1+([tex]\frac{sint}{1-cost}[/tex])2)=C

    -> (1-cost)(1+[tex]\frac{sin^{2}t}{(1-cost)^{2}}[/tex]) = C
    -> multiplied out i get (1-cost) +[tex]\frac{sin^{2}t}{1-cost}[/tex] = C
    -> [tex]\frac{1-2cost+cos^{2}t+sin^{2}t}{1-cost}[/tex] = C
    -> [tex]\frac{2(1-cost)}{1-cost}[/tex] = C
    -> 2 = C

    how does C = 2 answer "show that y=y(x) is the solution of the differential equation for any parameter a"?
     
  7. Dec 2, 2008 #6

    Dick

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    Re: Brachistochrone

    You missed an 'a'. I get 2a=C. That answers 1. For 2 if you have the substitution correct, then it looks like a u=tan(w) substitution.
     
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