How to solve this nonlinear diff eq of

avalg10o

avalg10o

Answered question

2022-04-01

How to solve this nonlinear diff eq of celestial mechanics?
(r˙)2=2μr+2h
Where mu and h are constants.
I have no idea how to solve it, maybe there is a trick I didn't know.
The only thing that came in mind is to integrate
dr2μr+2h=dt
but I don't think this is really a solution, since I don't know too how to evaluate the integral in terms of elementary functions.

Answer & Explanation

Sawyer Anthony

Sawyer Anthony

Beginner2022-04-02Added 10 answers

Step 1
Rewrite the integral as
12rμ+hr:dr
and use the substitution r=μhsinh2ρ
μ22h32sinh2ρ:dρ=μ22h3cosh2ρ1:dρ=μ22h3(sinhρcoshρρ)
Then undo the substitution with
sinhρ=hrμcoshρ=1+hrμ
Step 2
giving an equation
hrμ1+hrμsinh1(hrμ)=2h3μ(t+C)
This is of course only the case where an object keeps moving further and further away from a gravitational body because of your choice for r˙ to be strictly positive.

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