Calculus of variations problem and differential equation initial conditions

[catpants]

Calculus of variations problem. I want to make stationary the integral of (1+yy')^2 dx from 0 to 1. I know what the Euler-Lagrange differential equation turns out to be, but how do I interpret the limits of integration as initial conditions for the diff eq?

also, can i use laplace transforms to solve differential equations if I know a few initial conditions for y, but no initial conditions for y'?

Thanks!

 

[hunt_mat]

Okay, I will answer the second question first and then suggestions for the first.

The answer to this one is take a dufferential equation which you know the solution of and swap the intial condition for y' for something else and take laplace transforms of your equation and leave a general function (or possibly constant) in it's place and then go through as normal and see if you can't find out the constant using the other condition you have. I solve Lapalces's equation this way and it worked perfectly fine.

For the first question, can you solve te equation you get with some arbitrary constants in? If so plug this back into the integral, do the integral and then minimise the results in the usual way.