46 2. The First Variation In writing this chapter, we leaned heavily on Bolza (1973) and Sagan (1969). I encourage all students of the calculus of variations to read these two books. 2.5. Exercises 2.5.1. Euler’s approach. Using Euler’s approach from Section 2.2, determine polygonal approximations to the curve that minimizes 2 0 [(y )2 + 6x2y] dx (2.77) subject to y(0) = 2, y(2) = 4 (2.78) for n = 1, n = 2, and n = 3. Write down and solve the Euler– Lagrange equation for this problem. Compare your polygonal ap- proximations to your solution of the Euler–Lagrange equation. 2.5.2. Another lemma. Let M(x) ∈ C[a, b] be a continuous func- tion on the closed interval a ≤ x ≤ b that satisfies b a M(x) η (x) dx = 0 (2.79) for all η(x) ∈ C2[a, b] satisfying η(a) = η(b) = η (a) = η (b) = 0 . (2.80) Prove that M(x) = c0 + c1x (2.81) for suitable constants c0 and c1. What can you say about c0 and c1?

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