I've been trying this all day, hopefully I'm just being stupid. The full problem is to show that two simultaneous differential equations in x(t) and y(t) have a solution that is a circle, which I believe is the circle where mu is given and a is a constant depending on initial conditions. I've reduced the problem to showing
I can't show this, so if anyone could help with that I'd be really grateful.
I'm also confused as to where I'm going wrong here:
If and then the constant in the last expression above is zero (I hope that's not my mistake!). But then we can rearrange this (when x'(t) is non-zero and x isn't mu) to get which is false in most cases. Why can't I divide through?
Yeah, sorry. That's the last time I ask a question after midnight.
The original equations are
Where and in this particular case . To try and solve these, I divided the second by the first (or if x = mu, the first by the second) and cross multiplied to get
and we can rearrange and integrate this to get
which is of the form if and only if
The context is a spaceship orbiting a large planet. I just don't get why this last equation is true (which it must be, since the circle solves the original problem). Sorry for my vagueness before!