find the extremal of a function i need to find the extremal of the functional $\int I(y,y&#39;&#39;) dt$. Could anyone tell me the concepts of finding the extremal so that i can go about solving this one?

Update:

So my function was: G= $\int(ay+ $$\frac{1}{2}$$ b$y''^2$ )dt$
solving it using using the Euler–Lagrange equation as given in the link below, we get:

a+by''''=0

which on solving gives:

y=-(a(t^4))/(24b).

Am i right in solving this one?
Principally this should be related to the space on which the functional is defined. This depends on the exact form of $I$. However, if you forget these things, consider for $J(y)=\int I(y,y&#39;&#39;)$, the expression $J(u+\tau v)$, where $\tau$ is a parameter, and $u,v$ are functions. Then differentiate the expression which respect to $\tau$, and put $\tau=0$. Rewrite the expression, using partial integrations (this only works if the space you decide to define the functional on is nice enough, for example compactly supported functions on the domain you are working on). You should then find some expression of the form $\int B(u,u&#39;&#39;) v$. Since for a critical point this must be zero for all $v$, we conclude $B$ must be identically zero.
of course, the expression containing the terms that are unknown in the sense we trying a unknown rate of the function for the application with domain as the variable x. hence your answer in the predict of the Euler-Lagrange equation concept must needed the exact differentials in order to meet the bijection principle of a mapping.

ok, Asutosh sang K.M.P