Brachistochrone homework problem

[twalker40]

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, I am stuck at 1+y`<sup>2</sup>= C/y. i know your not supposed to subtract 1 to either side, so how am i supposed to isolate y` by itself?

 

[Dick]

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).

 

[twalker40]

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).

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

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

 

[Dick]

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.

 

[twalker40]

ok, I am 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), that's 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+(\frac{sint}{1-cost})²)=C

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

how does C = 2 answer "show that y=y(x) is the solution of the differential equation for any parameter a"?

 

[Dick]

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.