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Regular Points of Differential Equations of the Second Order
Excerpt from Regular Points of Differential Equations of the Second Order
We shall find it convenient to postpone the proof of this extremely important theorem until later, and to go on at once to the following, Definition A regular point of the equation (1) is a point at which p does not become infinite to an order higher than the first or q to an order higher than the second. It will be seen that all ordinary points and some singular points are included among the regular points. The other singular points at which p becomes infinite to an order higher than the first or q to an order higher than the second or at which both of these things happen we shall naturally speak of as irregular points. In the following sections we shall obtain in the neighborhood of regular points solutions of (1) in the form of series and in the course of this work we shall obtain as a special case the fundamental theorem above given concern ing non-singular points.
We shall find it convenient to postpone the proof of this extremely important theorem until later, and to go on at once to the following, Definition A regular point of the equation (1) is a point at which p does not become infinite to an order higher than the first or q to an order higher than the second. It will be seen that all ordinary points and some singular points are included among the regular points. The other singular points at which p becomes infinite to an order higher than the first or q to an order higher than the second or at which both of these things happen we shall naturally speak of as irregular points. In the following sections we shall obtain in the neighborhood of regular points solutions of (1) in the form of series and in the course of this work we shall obtain as a special case the fundamental theorem above given concern ing non-singular points.
25 pages, Paperback
First published August 22, 2015
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