FLuxions. differenntials and power series

[roadrunner]

Homework Statement

The firstthing I need to note is this is for a HISTORY of math course, so we have to use non modern techniques in most cases, some not. In other words, thequestion describes how to solve them. I'm also on a compyuter with a terrible keyboard so I'm doing my best.


1)
Derive the power series for the logarithm by beginning with dy=1/(x+1)dx and assuming that y is apowerseries in x with undetermined coeeficients, and solving simple equations to determine each coeefficient in turn.


2)solve the fluxonial equation x*/y*=2/x+3-x^2 by replacing x with x+1 and then using power series techniques

NOTE: x* and y* are simply x with a dot above it, indicting derivtive of x in the old notation

Homework Equations

1) 1/(x+1)=1-x+x^2-x^3+...

The Attempt at a Solution

1) Well I know that if i intergate this i get the corretc solution, but they didnt have our modern integrating notations at the time. (this is Leibniz) and it syas to determine coofeicients, so I have no idea where to start


2) i had no idea what to do here. I repce x with x+1 and got 2/(x+1) (whihc is 2 times the same expandion in 1) ) +3-(x^2+2x+1)

No idea wht to do after that, I don't even know what I'm suposed to solve for.

Thanks

 

[roadrunner]

on problem 1)

I said that y=a+bx+cx^2+dx^3...

that is, y is some power series of x,

then i took the derivative of that getting

dy/dx =b+2cx+3dx^2...

and let that equal the power series for 1/(x+1)

that is,

1-x+x^2-x^3+x^4=b+2cx+3dx^2

and get that b=1 2c=-1, this c=-1/2, and by the same logic d=1/3, e=1/4 etc etc.

Would that be correct?

 

[roadrunner]

for number 2, i thought about it more, and I'm thinking I need to solve to x*/y* in terms of a simple equation (like how 1-x+x^2-x^3+... becomes 1/(x+1)