Math Is Fun Forum

Jen_Jer_888

Member

Registered: 2011-07-28

Posts: 1

Power series solution for differential equation

The problem:

Solve the fluxional equation (y with a dot on top)/(x with a dot on top) = 2/x + 3 - x^2 by first replacing x by (x + 1) and then using power series techniques.

My feeble attempt at a solution:

First, I believe the fluxional (y with a dot on top)/(x with a dot on top) was just Newton's language and notation for the derivative dy/dx, so I rewrote the equation as dy/dx = 2/x + 3 - x^2. Then I replaced x by (x + 1) like it says, getting: dy/d(x+1) = 2/(x+1) + 3 - (x+1)^2
From there, I attempted to set it equal to the sigma series for the derivative of a power series, so: 2/(x+1) + 3 - (x+1)^2 = Sigma n*a_sub_n*(x+1)^(n-1) from n=1 to infinity = a_sub_1 + 2*a_sub_2*(x+1)+ 3*a_sub_3*(x+1)^2 + 4*a_sub_4*(x+1)^3 + ....
I don't know where to go from there. I'm not even sure how the substitution helps. Since there is no y or higher order derivative, I see no basis to compare series coefficients. I don't know what other power series techniques to employ.

Any help would be appreciated!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Power series solution for differential equation

Hi Jen_Jer_888;

You are correct with the problem, that is Newton's notation and probably the only dumb thing he ever did. In them days occasionally a fly would differentiate!

I know it is wrong, but today when I see someone write 2 / x + 3 they usually mean 2 / (x+3).

Question: What is the purpose of substituting x +1 for x?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Power series solution for differential equation

Hi all,

If the question is to solve the differential equaation, what has power series got to do with it?!

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."