4. (i) Show that applying the Euler Lagrange equation to the functional (* – +249()) dr leads to the second order differ

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4 I Show That Applying The Euler Lagrange Equation To The Functional 249 Dr Leads To The Second Order Differ 1 (23.82 KiB) Viewed 33 times

4. (i) Show that applying the Euler Lagrange equation to the functional (* – +249()) dr leads to the second order differential equation y"+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 cosx +vsin x for two new variables u and v, and assume the convenient relationship ucose + o'sin x = 0 (CR2) between u and v, we get the pair of first order equations '= -9(x) sin r and d = g(x) cose. Why is (CR2) a convenient relationship? (iii) Choose a nonconstant function g(x) and find , , and y.