How to eliminate an independent variable from two coupled differential equations?

Question

I want to solve following differential equations using mathematica (I know the solution as it is easy to compute): $$\dot{r}^2+r^2\dot{u}^2=2\kappa^2\frac{1}{r}+h,$$ $$r^2\dot{u}=c.$$ To computer the solution we substitute second equation in the first and then eliminate the time variable from these equations using, $$\dot{r}=\frac{dr}{du}\frac{du}{dt}=\frac{dr}{du}cr^{-2}.$$ Substituting this we get $$\left(\frac{dr}{du}\right)^2\frac{c^2}{r^4}+r^2\frac{c^2}{r^4}=2\kappa^2\frac{1}{r}+h.$$ Can someone please show me how to do this using mathematica?

I have tried following things, but that didn't work:

expr = r'[t]^2 + r[t]^2 u'[t]^2 - 2 κ^2/r[t] - h

expr1 = expr /. r -> (r[u[#]] &)

expr2 = Eliminate[{expr1 == 0, u'[t] == c r[u[t]]^-2}, u'[t]]

expr3 = expr2 /. {r'[u[t]] -> c r[u[t]]^-2 r'[u[t], u[t]], u'[t] -> c r[u[t]]^-2}

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@michael-e2 I know about DSolve and NDSolve. But to solve this question further I have to do another change of variable. Therefore I'm not using them. — siddhesh, Mar 07 '16 at 13:20
This is related Working with a system of differential equations that cannot be solved explicitly — Artes, Mar 07 '16 at 13:27
This is also related:http://mathematica.stackexchange.com/q/80241/1871 — xzczd, Mar 07 '16 at 13:38

score 2 · Accepted Answer · answered Mar 07 '16 at 13:09

What do you want us to show you is not quite clear. I understand your question is the following way: "How to make within Mma the same steps you have done on the paper?" Is that right? If yes, the answer is as follows.

This is the left-hand part of your first equation:

expr1 = r'[t]^2 + r[t]^2 u'[t]^2 - 2 \[Kappa]^2/r[t] - h

Let us substitute your second expression into it. You might use Eliminate, but in this simple case substitution is easier:

    expr2 = expr1 /. u'[t] -> c/r[t]^2
(*  -h + c^2/r[t]^2 - (2 \[Kappa]^2)/r[t] + Derivative[1][r][t]^2  *)

Now, you replace the derivative by your third equation, and r[t] replace by r[u]. Let us do it:

 expr3 = expr2 /. {r[t] -> r[u], r'[t] -> r'[u]*c/r[u]^2}

(*  -h + c^2/r[u]^2 - (2 \[Kappa]^2)/r[u] + (c^2 Derivative[1][r][u]^2)/
 r[u]^4  *)

That's all. Here is your equation:

Then you can solve it using DSolve. You could have done this from the very beginning without transformations though. The solution is not very comfortable, but one can use it in several ways.

Have fun!

Thanks @Alexei. That was very simple indeed. – siddhesh Mar 07 '16 at 14:18 — siddhesh, Mar 07 '16 at 14:18

How to eliminate an independent variable from two coupled differential equations?

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