Theory of Differential Equations volume 2

Cover Theory of Differential Equations volume 2
Theory of Differential Equations volume 2
Forsyth, Andrew Russell, 1858-1942
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When the equation is in this form, it is known that a finite number of operations is sufficient to transform it to the preceding case, these operations being transformations of the type The last of the dependent variables v thus arising is that which occurs in the preceding investigation ; it has been expressed as an absolutely converging power-series. When the successive substi tutions are carried out, so as to give the initial dependent variable, the result manifestly is to give a regular fun...ction of t, that vanishes with t and, owing to its construction, satisfies the differential equation. Hence we have the theorem : F. II. 10 146 NON-REGULAR INTEGRALS OF THE [64.
The differential equation dv t-j7 = at where the coefficient a is not a positive integer, possesses a regular integral which vanishes when t = and exists over a finite part of the region of existence of the function fa(v, t).
The argument of 12 may be applied, or a proof similar to that in 13 may be constructed, to shew that the regular integral thus obtained is the only regular integral which vanishes with t and satisfies the equation.


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