Differential Equation & MacLaurin Series for Newton’s Second Law

Question

I am currently working with a differential equation, where I think I need to take the derivative of $ma$ (corrected as per comment). I am trying to write $F = ma$ as a MacLaurin series and eventually set it in terms of $mddot x(t)$. The problem is that I am not sure if I should write my MacLaurin series in terms of $x$ also or use a. Also, I am not very sure if you take the derivative of $a$ if you would have to use Chain Rule or if you could simply take the third derivative of position. Any hints would be appreciated.

Daddy Kropotkin · Accepted Answer

As I understand the problem statement, you want to start with Newton's 2nd law,

$F$ = ma

for some general Force $F$. Let's suppose we're talking about the 1-dimensional displacement $x$ of a particle with mass $m$ at time $t$.
Writing $F(x)$ means

$F(x) = ma = m*left(frac{d^2 x}{dt^2}right)$.

which we shall call Eq. (1). It seems you've gotten that far on your own.
Now, write down the Taylor series (it's a "Maclaurin series" since the origin of the particle's displacement is zero, i.e. the series is expanded about zero) for a function $F(x)$. Don't write terms higher than the "second-order" derivative. Call this Eq. (2).
Now, set Eq.'s (1) and (2) equal to each other (i.e. substitute in for $F(x)$) and see what you get.

Differential Equation & MacLaurin Series for Newton’s Second Law

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