The roots of polynomial equations in one variable can be shown easily to satisfy 2 independent linear partial differential equations with respect the equations coefficients. The discriminant also satisfies 2 similar independent LPDE’s.

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.

An inequality for the area of a triangle A is derived in terms of the perimeter p. This inequality is related to the Weitzenbock inequality and is an improvement on it. The method used also throws up a second triangle inequality. A further link to the Hadwiger-Finsler inequality is also shown. The Ono inequality for acute angled triangles is also proved.

There are a number of inequalities that can be easily derived for the general cuboid. These inequalities connect the Perimeter P, Surface area S and the Volume V. The proof of one of these I have seen elsewhere, but I have not seen all of these published together. These can be easily applied to the general triangle via Heron’s formula. See the ‘Knol’ on triangle inequalities for the applications.

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.