894 



THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



Of the many methods which have been devised for finding solutions of 

 Mathieu's differential equation, the one which is conceptually simplest is 

 that due to Bruns. This method can be described as follows: 



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Fig. 3 — Stability diagram for Mathieu's differential equation 



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Under the transformation 



exp 



J en 



the Mathieu equation (3') goes over into the Riccati equation 

 d<p 



dz 



+ <p' + el + IBi cos 2z = 0. 



(5) 



We seek a solution of this equation in the form of a power series in the 

 parameter ^i, say 



ip{z) = <po(z) + dm{z) + el <p2(z) + ... , 



and we easily find that the functions ^o(z), ^iW, ^2(2), ... must satisfy the 

 differential equations 



<po + (Po + eo = 0, 



<pi + 2<po(pi + 2 cos 2z = 0, 



(p2 + 2(pQ(p2 + ^1 = 0, 



(pi -h 2<po</'3 + 2<pi<p2 = 0, 



<PA -\- 2<P(]fPA + 2(pi^3 + <^2 = 0, 



(6) 



