# CoV16

From Example Problems

Minimize

Let denote the integrand in the above. The Euler equation (see CoV4) for this problem is

The characteristic equation for this differential equation is
which factors to ; so the roots are .
Thus, the general solution of the differential equation is

The conditions at infinity require that . Thus,

Applying the remaining conditions,

so and so the desired solution is

Satisfying the Euler equation is a *necessary* condition for the given solution to minimize
; proving that it actually does minimize is a lot more
difficult (and will not be dealt with here).