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Homework Help: FLuxions. differenntials and power series

  1. Apr 5, 2010 #1
    1. The problem statement, all variables and given/known data

    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


    2. Relevant equations

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


    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
     
  2. jcsd
  3. Apr 5, 2010 #2
    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?
     
    Last edited: Apr 5, 2010
  4. Apr 5, 2010 #3
    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)
     
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