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).