Solve problem of Newtonian Dynamics?

Questions
,
1

I want to solve this differential equation (attachment) in mathematica. How should I do this?
Many Thanks.

1.pdf (85.5 KB)

2

did you try DSolve?

3

Change to polar coordinates.

Use (3/2) for the power instead of 1.5 — it’s generally better to use an exact number like the Rational number 3/2 than a floating-point Real number like 1.5 which is treated as an approximate number. Sometimes it doesn’t matter; sometimes it does.

4

This seems to be THE classical problem of Newtonian Dynamics.

With r^2 = x^2 +y^2 (change to polar coordinates as Michael proposed) my (old) textbook gives

t - t0 = Integrate[1/(Sqrt[2/m (Energy - U[r] - C^2/(2 m r^2))]), r]

and

phi = Integrate[ C / ( m r[ tt ]^2 ), {tt, 0 ,t }]

Where Energy is the total Energy of the body and C its angular momentum.

As far as I remember at least one of the integrals is an ellipitical integral without closed solution, so NIntegrate would be appropriate.

But then of course one could directly use NDSolve.