I was working problem 3.5 out of Hartle's "Gravity" and have come to a bit of confusion. The problem states that the action for some system is given by the integral from 0 to T of d(x(t)/dt squared plus x(t) squared. My next step was to take the Lagrangian and set it equal to zero. Doing this, I found that the second derivative of x was equal to negative x, which is suggestive of exponential behavior. I then assumed the form e^x and solved for the initial conditions listed (x(0)=0, and x(T)=1). Is this all sound? Something about it seems off, but I'm not sure where it is that I may have taken a misstep. Thanks!