Let us remark that the looked for differential equation can be written under the form
involving solutions of the form :
otherwise said with cartesian equation :
where $a$ is any real.
Setting $\alpha=\arccos(x)$ in (3), we get the equivalent parametric equations :
in which we recognize that we are working with elliptical arcs, as shown on the graphical representation of the family of curves $C_a$ displayed below (we have to pay attention to the domains of variables $x$ and $y$ : in general we will not have the whole ellipses as solutions but arcs of them).
From (4), it is easy to establish a connection with relationship:
(in the spirit of the solution given by @CY Aries).
Fig. 1:Curves $C_a$ for $a=-\pi$ to $a=\pi$ with step $\pi/8$ (progressively changing from blue to red). The curves are elliptical arcs with two degenerate cases (straight lines).