Question

Is it possible to solve the differential equation mr''(t)=-GMmr^-2 for r(t), t?

Answer 1

@wio

Answer 2

I can't seem to even get a numeric solution on wolfram alpha or mathematica because I just can't use the syntax correctly

Answer 3

Well, I know the equation exists and it is a real valued function.

Answer 4

function^

Answer 5

I think of reduction of order.

Answer 6

But I don't know it is an elementary function.

Answer 7

It looks separable.

Answer 8

to reduce the order from 2 to 1 by letting u = r' t

Answer 9

But then again, it isn't first order.

Answer 10

sure, that's why I use that method

Answer 11

\[
m\frac{d^2r}{(dt)^2}=G\frac{Mm}{r^2} \\
r^2\frac{d^2r}{(dt)^2}=GM
\]

Answer 12

minus sign from the RHS, but then?

Answer 13

What happens if you do it your way?

Answer 14

http://www.wolframalpha.com/input/?i=y%28x%29^2*y%27%27%28x%29%3Dc
for some reason wolfram alpha won't give me an answer if I put a negative sign...

Answer 15

I didn't do, I tagged you , right? because I am not sure about it. ok, let me try

Answer 16

Maybe intergration by parts @Loser66

Answer 17

to me, I let u = r'(t)
u' = r"(t) = \(\dfrac{-GM}{r^2}\)
then du =\(\dfrac{-GM}{r^2}\)dr
integral both sides to get u

Answer 18

oh, to make life harder, I'm given r(0)=149,600,000,000, r'(0)=0, G=grav constant, M equals mass of the sun

Answer 19

@inkyvoyd numbers doesn't matter, plug them at the last step

Answer 20

@Loser66 if numbers don't matter then why is it that wolfram gets all flustered when I give it the boundary values?

Answer 21

@Loser66 Wait are you saying \[
\frac{du}{dr}=\frac{d^2r}{(dt)^2}
\]Or\[
\frac{du}{dt}=\frac{d^2r}{(dt)^2}
\]

Answer 22

Pick one.

Answer 23

last one

Answer 24

That would mean: \[
du=u'dt \neq u'dr
\]

Answer 25

oh yea, you are right,

Answer 26

surrender, It's toooooough.

Answer 27

uhm, this might not be the best time to mention this, but it just so happens I meant BVP and not Diff eq in my question

Answer 28

Hmm, It makes me want to guess a trig function like \(\sin(t)\)

Answer 29

But that doesn't really make sense does it.

Answer 30

I GOT IT.

Answer 31

the numerical solution I mean

Answer 32

How?

Answer 33

wasn't using syntax correctly. But the solution is as expected (the problem I gave was what were to happen if one stopped the earth's orbit and watched as it hit the sun)

Answer 34

Yeah it is the position of anything heading towards another mass with gravity

Answer 35

It's not hard to see it sort of is negative exponential, but

Answer 36

It's plotting on mathematica for me, but I don't know whether or not it's not converging the way i want it to or I'm just bad