4. (i) Show that applying the Euler-Lagrange equation to the functional † (1912 - y2 +299(x) dx leads to the second orde

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4 I Show That Applying The Euler Lagrange Equation To The Functional 1912 Y2 299 X Dx Leads To The Second Orde 1 (23.8 KiB) Viewed 25 times

4. (i) Show that applying the Euler-Lagrange equation to the functional † (1912 - y2 +299(x) dx leads to the second order differential equation x"+y=9(). (ii) The rest of this problem recalls the variation of parameters method of solving this equation. Show that if we set y = u cos2x+usin.x for two new variables u and v, and assume the convenient relationship u'cos x +v'sinr = 0 (CR2) between u and v, we get the pair of first order equations ' = -9(x) sin.x and ' = 9(x) cos r. Why is (CR2) a convenient relationship? (iii) Choose a nonconstant function g(x) and find , , and y.