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Gateaux differential and Euler-Lagrange equation

Hi! Can anyone help guide me through how to find the Gˆateaux differential and use

it to derive the associated Euler–Lagrange equation for the following functional? Satisfying and boundary conditions also.

Attachment 29892

Im really sorry, I was having so many problems with trying to create that in Latex, so I have attached a thumbnail showing the function needed, click it and theres a clear version of the function!

Many thanks

Re: Gateaux differential and Euler-Lagrange equation

Sorry, I have no idea how to do your problem but I've been learning latex (very slowly) so I thought I would show you that.

(I hope that you don't mind)

$\displaystyle S[y]=\int_1^2dx(3y'^2-2y^2),\;\;y=(1)\;\;y(2)=3.|$