1.3.3  The Fundamental Theorem of Calculus of Variations If we introduce the function g(x) by then Lemma: If    then g(x)=0 in [a,b], where g(x) is continuous on [a,b]. This famous lemma is known as the fundamental theorem of calculus of variations. Now, returning back to our problem, the use of the previous lemma yield the following two-point boundary value problem to determine the optimal solution     (1) This is called as the "Euler" differential equation associated with the functional I(y). In general, it is a nonlinear, 2nd order, ordinary differential equation and hard to solve analytically.