Theory of Differential Equation. (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ... function of degree m--y, shall have m--y--8 linearly independent regular integrals, is that JV shall be a product of the form QMD, where the indicial functions of, i/, D are of degrees 5, 0, m--y--b respectively, and D is of order m--y--b. Is there any limitation upon the order of M1 (Cayley.) Ex. 5. Shew that an equation QD = 0 has at least as many regular integrals as Z) = 0, and not more than Q = 0 and Z) = 0 together; and that, if all the integrals of D=0 are regular, then QD--0 has as many regular integrals as Q = 0 and D = 0 together. Hence (or otherwise) shew that, if an equation P=0 has all its integrals regular, then P can be resolved into a product of operators, each of the first order and such that, equated to zero, it has a regular integral. Is this resolution unique (Frobenius.) 78. In the two extreme cases, first, where the degree of the indicial function is equal to the order of the equation, and second, where its degree is zero, the number of regular integrals is equal to that degree. The preceding proposition shews that, in the intermediate cases, the degree merely gives an upper limit for the number of regular integrals. It is natural to enquire whether the number can fall below that upper limit. As a matter of fact, it is possible to construct equations, the number of whose regular integrals is less than the degree of the indicial function. Taking only the simplest case leading to equations of the second order, consider the two equations of the first order; and form the equation ax dec which manifestly is of the second order, say d2y dy where _ h-dh __dk k dh h doc' doc h doc' If we can arrange so that oc = 0 is a pole of p of order n, where n 2, then oc = 0 in general will be a pole of q of order n + 1; and the indicial function ...