Undetermined Coefficients. i6i 



A=i, 

 2A„A,= — I, 



2AA+A;=o, 

 2A^A^4-2A A^=o, 



2A^A^+2A A^+A/=o, 

 2A^A^+2A A^+2A^A^=o, 

 2A.A,+ 2A,A^4-2A,^A_,+A/=o, 

 etc. 

 From these we get 



A=i, 



2A^ 



A — ^-^ 



~ 2a; 



A 2 A. A, 



A 2A,A3+a; 



2\ ' 



A 2A.A,+ 2A ,A3 



5 2A„ 



_ 2A Aj+2A,A. + A,' 



A,_ ^2-A---- ■• ■ 



etc. 

 From these the law of the series can be seen. 

 Taking these equations in order, we find the numerical value 

 of the undetermined coefficients to be as follows : 



Ao=i» A^=— J, A^=— I", A^=— y\, A_^= — ify, 



A — 7 A 21 



^5— 2"5tt' ^6— TT2T- 



Making these substitutions in the assumed development, we 

 obtain 



, __ X x^ x^ 5jr' yx^ 2i.f^ 



2 8 16 128 256 1024 



13. Examples. 



1. Develope 1+ 2Jf— • 



2. Develope \J x-\-x:\ 



J. Develope (\-\-x)^. 

 A— 20 



