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    The Euler-Lagrange Equation for a Variational Problem

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    Determine the Euler-Lagrange equation for the variational problem arising from the integral of F(x, u, u_x) = x u_x and discuss what conclusions can be drawn from the result. Here, u_x denotes the partial derivative of the dependant variable u with respect to x.

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    Solution Summary

    The Euler-Lagrange equation for an ill-posed variational problem is considered. It is shown that there is no solution to the Euler-Lagrange equation in this case because the corresponding variational problem has no extrema.