# Finding the functional extremum

## Homework Statement

I have been given a functional
$$S[x(t)]= \int_0^T \Big[ \Big(\frac {dx(t)}{dt}\Big)^{2} + x^{2}(t)\Big] dt$$
I need a curve satisfying x(o)=0 and x(T)=1,
which makes S[x(t)] an extremum

## Homework Equations

Now I know about action being
$$S[x(t)]= \int_t^{t'} L(\dot x, x) dt$$
and in this equation $$L= \Big(\frac {dx(t)}{dt}\Big)^{2} + x^{2}(t)$$

## The Attempt at a Solution

Is there any other way I can express Lagrangian to fit in this equation and hence I can do the integral? and is there any general solution for the lagrangian?