This is more an extended comment as a solution. For convenience, begin from the ODE in the form given in the Wikipedia reference cited in the question.
y'[x]^2 == 2 r/y[x] - 1
The OP wishes DSolve
to provide the parametric answer,
{y[t] -> r - r Cos[t], x[t] -> r (t - Sin[t])}
However, this result is not unique. For instance, replacing t
by a t
, where a
is any real constant, also is a valid parametric solution. And, there are infinitely more equivalent parametric solutions. Obtaining the desired solution requires specifying an additional relationship.
Here is a simple illustration. Rewrite the ODE in terms of a parameter t
.
y'[t]^2 == x'[t]^2 (2 r/y[t] - 1)
This ODE cannot, of course, be solved without a second relationship among x
, y
, and t
. For instance, specify
x'[t] == y[t]
(Other second equations are possible but yield different parametric representations.) Now solve the set of ODEs, impose the boundary conditions that x
and y
vanish at t == 0
, and simplify
FullSimplify[First@DSolve[{y'[t]^2 == x'[t]^2 (2 r/y[t] - 1), x'[t] == y[t],
x[0] == 0}, {x[t], y[t]}, t] /. C[1] -> 0]
(* {y[t] -> r - r Cos[t], x[t] -> r (t - Sin[t])} *)
as desired. This example is meant to show that DSolve
is capable with assistance of producing parametric solutions. It is not, however, capable of producing any particular parametric solution that someone might desire without very particular assistance.
DSolve[]
is capable of parametric solutions yet. Here's another simple example (the equiangular spiral):DSolve[{y'[x] == (x + y[x])/(x - y[x]), y[1] == 0}, y, x]
. $\endgroup$