I am trying to find a function y(x) which makes

ò_x1^x₂((y ' )²+Öy) dx stationary.

However, applying the Euler-Lagrance equation this ends up as

(d/dx)(2y ' )-1/2Öy=0

or

y ' ' =1/(4×Öy).

I have found one solution to this by guesswork (y=(3x/4)^4/3) but I really need a general solution. I don't know any methods for non-linear second-order equations. Can anybody help?

David

David,

Here's a nifty way of solving certain higher order diff. eqns, one's that don't have the independent variable x explicitly:

Let p=dy/dx.
Then y" = dp/dx = (dp/dy)(dy/dx) = p dp/dy

So your eqn becomes p dp/dy = 1/(4*sqrt(y)) which is nice and separable and gives you

p² = sqrt(y) + c, or

dy/dx = sqrt(sqrt(y) + c).

This too is separable, and can be solved for x in terms of y. When c=0, you get your solution above (slightly generalised): y(x) = (3[x-x₀ ]/4)^4/3 . For general c, you can solve it by using the substitution v=(sqrt(y) + c). I didn't work through the algebra fully, but I'm not convinced you'll be able to write y in terms of x in this general solution.

Pras

Thanks.