Nonlinear ODE from optimization problem

I'm working through the Brachistochrone Problem but I get stuck at the differential equation:

$\displaystyle (1+y'^2)y=K^2$ (eq. 13)

The solution is supposed to be in the form of two parametric functions $\displaystyle x(\theta)$ and $\displaystyle y(\theta)$, so I tried switching to polar coordinates and manipulating the equation in various ways. But that lead me to some hairy integrals, and I want to find out if I'm on the right track before I continue.

What's the best approach here?

Re: Nonlinear ODE from optimization problem

Quote:

Originally Posted by

**TwoPlusTwo** I'm working through the

Brachistochrone Problem but I get stuck at the differential equation:

$\displaystyle (1+y'^2)y=K^2$ (eq. 13)

The solution is supposed to be in the form of two parametric functions $\displaystyle x(\theta)$ and $\displaystyle y(\theta)$, so I tried switching to polar coordinates and manipulating the equation in various ways. But that lead me to some hairy integrals, and I want to find out if I'm on the right track before I continue.

What's the best approach here?

Hint: The equation is separable.

-Dan

Re: Nonlinear ODE from optimization problem

Quote:

Originally Posted by

**topsquark** Hint: The equation is separable.

-Dan

Thanks. So after separating I get:

$\displaystyle x=\int_0^y\sqrt{\frac{y}{K^2-y}}dy$

But this feels like another dead end. Even if I could solve the integral, I'm not sure I would know what to do with the result...

Re: Nonlinear ODE from optimization problem

Look at the article that you posted earlier. Take a look at equations 30 - 33. Specifically take a look at the substitution in equation 31. That's the key to integrating the problem, but I'm afraid that I can't make it work. If you like you can work backward and verify that $\displaystyle x( \theta) \text{ and } y( \theta )$ satisfy the differential equation. (It's fussy about the details, but isn't *too* hard overall.) That's as far as I can go with it.

-Dan

Re: Nonlinear ODE from optimization problem

Quote:

Originally Posted by

**topsquark** Look at the article that you posted earlier. Take a look at equations 30 - 33. Specifically take a look at the substitution in equation 31. That's the key to integrating the problem, but I'm afraid that I can't make it work. If you like you can work backward and verify that $\displaystyle x( \theta) \text{ and } y( \theta )$ satisfy the differential equation. (It's fussy about the details, but isn't *too* hard overall.) That's as far as I can go with it.

-Dan

Thanks, I will give it a try. It could be that when people first solved this, they just guessed at a solution that made sense (a cycloid), and then checked to see if it worked.