By F. M. Arscott
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Additional resources for Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions
COROLLARY 3 . If Mathieu's equation possesses a basically-periodic solution, the second solution is not periodic at all. F o r by Corollary 2, the second solution then has the form ζ X (periodic function)+periodic function, and this cannot be p e r i o d i c * COROLLARY 4 . Solutions 2π are basically-periodic. of Mathieu's equation with period F o r solutions with period 2π can exist only (Theorem 5, Cor. e. when there exists one basically-periodic solution. But in this case the other solution is n o t periodic, so the solution with period 2π must be the basically-periodic solution.
We only consider Mathieu's equa tion, but several theorems apply to other singly-periodic equa tions also, and more have close analogies. The standard form of Mathieu's equation is taken, as before, to be — + (a-2qcos2z)w dz^ = 0, (1) and the result of writing iz for ζ is the "modified Mathieu equation". d^w - ( a - 2 ^ c o s h 2 z ) = 0. (2) dz2 Since (1) and (2) have no finite singularities, every solution is an integral function of z. The first theorem is simple, but very useful: • This chapter develops the theoiy of Mathieu's equation without finding explicit solutions; it can safely be omitted by the reader who is interested only in applications and is prepared to take the theory for granted.
W¿n) W i ( z ) - ) V 3 ( j r ) >vi(z) (by ( 1 ) ) = 0. Hence w ( z ) has period π a n d the result is established for (i) with the upper sign; the other three results m a y be proved similarly. COROLLARY 3 . If Mathieu's equation possesses a basically-periodic solution, the second solution is not periodic at all. F o r by Corollary 2, the second solution then has the form ζ X (periodic function)+periodic function, and this cannot be p e r i o d i c * COROLLARY 4 .