5 Differential Equations

This section treats equations for an unknown function, taken up where a first course leaves off.

Three strands run through it. The first is systems: several unknown functions tied together by several equations, solved here by successive elimination until one equation in one unknown remains. The second is the operator method, in which differentiation is written as \(D\) and the equation is manipulated much as a polynomial would be — a notation that turns some intricate problems into algebra.

The third is what to do when the coefficients are not constant and no elementary method applies. There the solution is sought as a power series, and the section ends with the method of Frobenius, which extends that idea to equations with a singular point — the case that arises in almost every application where the geometry has a centre.

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