By Bartolucci D., Leoni G., Orsina L.

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**Example text**

Evaluate these coefficients. This is the only one of Newton's techniques that will be discussed in this section by means of several examples that do not involve second or higher order derivatives. The general outline of his method is the same for all of the examples and seems eminently reasonable. In fact, however, several of the steps need justifications that will only be provided in Chapters 13 and 14. The variable y 35 36 NEWTON'S CALCULUS (PART 2) in the proposed differential equation is assumed to have an (unknown) power series expansion in terms of x.

Consequently, 2 - M2 T-5 = (2 1 + M - 3M^ M 2 )(1 - M + 4M 2 - 7M 3 + 19M 4 ■ · ·) = 2 - 2M + 7M2 - 13M 3 + 34M 4 Newton's Rule III also refers to analogs of decimal methods for the extraction of roots and the solution of equations. There was at that time a AREAS AND INFINITE SERIES 29 well-known procedure for extracting square (and higher order) roots that was commonly taught in high schools until very recently and whose details were so onerous as to make the long-division process seem like child's play.

1 + 1/3- - 1 2—x ( 2 ■ 1 M K 2 1+ - ■ f X2 9 " We now restate and comment on the three steps in each iteration of Newton's procedure. THE SOL UTION OF AL GEBRAIC EQUATIONS 43 Step 1: Extract the xfc-level equation out of Fk(x,tk) = 0. If the term tk in the equation Fk(x, tk) =0 were replaced by its power series expansion of Equation (5), each summand would have xk as a factor. The reason for this is that the Step 3 substitutions of iterations 0 , 1 , 2 , . . , k — 1 have eliminated all the terms of degrees 0 , 1 , 2 , .

### Semilinear equations with exponential nonlinearity and measure data by Bartolucci D., Leoni G., Orsina L.

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