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In the long-standing inverse problem of the calculus of variations one is asked to find a Lagrangian and a multiplier so that a given differential equation, after being multiplied with the multiplier, becomes the Euler–Lagrange equation for the Lagrangian. An answer to this problem for the case of a scalar ordinary differential equation of order \(2n, n\ge 2,\) is proposed.

The multiplier problem of the calculus of variations for scalar ordinary differential equations