A method is described for arriving at a power series expression for a real root of a Bring-Jerrard form quintic equation. The method relies on showing that any solution of the quintic must also satisfy a 4th order LDE. The 4th order LDE is easily solvable to give a power series expression for the solution.
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A method is described for solving cubic equations. The roots of a cubic equation are treated as smooth functions of the equations’ coefficients and shown to be solutions to a second order linear differential equation which is easily solvable.