CHAPTER XII. 



UNDETERMINED COEFFICENTS. 



x^ a" . ' 



1. We know that =^x-\-a, and if we intesrralize this we 



x—a ^ 



obtain an equation of the second degree, but an equation of a 

 diiferent kind from those treated in Chapters IV and V, for the 

 equations previously treated under the name quadratics were 

 shown in Chapter V, Art. 9 to have two roots, and only two; that 

 is, it was shown that there were two and only two values of the 

 unknown quantity which would satisfy the equation ; but here 

 we have an equation of the second degree which can be satisfied 

 by any value whatever of x. 



The reason is that when the equation is in the integral form we 

 have exactly the same function of x on each side of the sign of 

 equality. 



2. Theorem. If tivo functions of x of the n th degree, 

 A^+Aa-H- . . . -j-A„x" and B^-^Bx+ . . . -I-B„x" , a?-e e^ua/ 

 for every value of x, theti the coeffLcients of like powers of x on the two 

 sides of the sign of equality are equal each to each. 



If the two functions are equal for every value of x, we have 

 A^ + Aa-H- . . . -f A„x"=B„+B,.r+ . . . -VB„x\ (i) 

 and since this equation is true for any value of x, we may con- 

 sider X as a variable, varying in any way we pleavSe. 



Then if we consider x to approach a limit, each side of the 

 equation is a variable which approaches a limit, and we have two 

 variables which are always equal, aiid each approaches a limit, 

 hence by Chapter XI, Art. 7 the limits are equal. Suppose x to 

 approach zero as a limit then 



limit of A„+A x-h . . . +A„j»:"=A^ 

 and limit of B^ + Bx^ . .. . +B,-r'=B^, 



hence A^=B„ by Chap. XI, Art. 7. (2) 



Subtracting A^ from the left side and B^ from the right side of 

 (i) we get 



Kx^-Ax^-{- . . . +A„.i-"=B,.r-f BX+ . • . +B„.v' (2,) 

 Divide (2,) by x and we have 



A. + A^.r+ . . . -f A,..v'— =B^ + B_A-+ . . . +B.X'-' (a) 



A-19 



