# Differential Equation & MacLaurin Series for Newton’s Second Law

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 $$m\ddot x(t)$$. 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.

• I am really confused because if you plug in x=0 to solve for the constants, you get just zero for everything! And that can’t be right. Nov 1 '20 at 4:40
• Can you post the exact problem, you are talking about? Nov 1 '20 at 4:50
• For the derivative of a^2 you certainly need to use chain rule the third derivative of position wouldn't work. Apart from that I am not very familiar with the MacLaurin series (haven't studied that yet) so I can't help on that Nov 1 '20 at 4:51
• Here is the start of the question: (a) Using Newton’s second law, write a differential equation for the position x(t) of an object experiencing a Force F(x). (b) Any oscillator moves back and forth across some equilibrium point. For simplicity you can always define that equilibrium point to be at x=0, in which case it is natural to expand F(x) in a MacLaurin series. Write a second-order MacLaurin series for F(x) and plug this into the equation you wrote down in part (a). The result should be a diff. eq. with x”(t) on the left and three terms on the right. Please no answer but hint(s) -thanks! Nov 1 '20 at 5:06
• $F=ma^2$ cannot be correct because it is dimensionally inconsistent. Therefore you have made a mistake somewhere in getting to this equation. Checking dimensions is a very useful way to find mistakes. Nov 1 '20 at 5:24

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.

• Thanks very much! This helped a lot! Nov 2 '20 at 7:37
• @Yelena My pleasure! If you've solved your problem, please accept my answer as the "correct" one. But you can still ask questions here after doing so. Thanks! Nov 3 '20 at 13:09