1. ## Brachistochrone ODE

Hey guys,
I'm a bit rusty on differential equations and today I was helping a friend solving ODE when we stumbled on this one:

$\displaystyle (1+ (y')^2)y = k^2$

The problem asked to solve for $\displaystyle y'$. Can you point me in the right direction? Thanks!

2. ## Re: Brachistochrone ODE

suppose y'=p we get
$\displaystyle (1+p^2)y=k^2$ gives $\displaystyle y=\frac{k^2}{1+p^2}$
now differentiating wrt x and putting $\displaystyle \frac{dy}{dx}=p$solve the equation in p and x

3. ## Re: Brachistochrone ODE

Originally Posted by TomAlso
Hey guys,
I'm a bit rusty on differential equations and today I was helping a friend solving ODE when we stumbled on this one:

$\displaystyle (1+ (y')^2)y = k^2$

The problem asked to solve for $\displaystyle y'$. Can you point me in the right direction? Thanks!
With simple steps You obtain the pair of ODE...

$\displaystyle y^{'} = \pm \sqrt{\frac{k^{2}}{y}-1}$ (1)

... and in each of them the variables can be separated...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$