How to Find the Functional Extremum for Given Boundary Conditions?

[tfhub]

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?

 

[Orodruin]

Why do you want to do the integral? What is wrong with using the Euler-Lagrange equations? This will give you a differential equation that your solution must satisfy to be an extremum the functional.