Let denote the integrand in the above. The Euler equation (see CoV4) for this problem is
The characteristic equation for this differential equation is m4 − 2m2 + 1 = 0 which factors to (m − 1)2(m + 1)2 = 0; so the roots are m = 1,1, − 1, − 1. 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).