Solving for $x$ in this simple differential equation?

Question

$\dfrac{dx}{dt}=2\dfrac{\sqrt{2g(\sin c- \sin x)}}{\sqrt{l}}$. $g$, $c$, and $l$ are all constants. Is there a way to solve for $x$ in terms of $t$ here? Once I did separation of variables and plugged in the integral into wolframalpha I got a pretty horrendous integral on the side with $x$. I was wondering if it could perhaps be simplified especially when you solve for $x$ in terms of $t$. Thanks

Answer 1

If it's OK for you to use the Taylor series approximation of $\sin(x)$ as $\sin(x) \approx x$ (for small x), then you can rewrite your equation as

$\dfrac{dx}{dt}=2\dfrac{\sqrt{A - 2gx}}{\sqrt{l}}$

where $A = 2g\sin(c)$.

You then have a more straightforward separable equation with the solution

$x(t) = \frac{Al - g^2(c_1 +2t)^2}{2gl}$.

This is called the "small angle approximation."