# Solving for $x$ in this simple differential equation?

$\dfrac{dx}{dt}=2\dfrac{\sqrt{2g(\sin c- \sin x)}}{\sqrt{l}}$. $g$, $c$, 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

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Nonlinear, inhomogeneous, implicit ODE? Maybe it's doable, but my gut feeling says NO. –  Kaster May 2 '13 at 2:14
Hold on, are you trying to solve pendulum problem? There's no closed form for that in elementary functions. –  Kaster May 2 '13 at 2:20
Yup lol how did you know? Did you try it once yourself too? –  Ovi May 2 '13 at 2:21
I know because the ODE looks familiar. I learned being in high school that this ODE cannot be solved in elementary functions. You can use elliptic integral though. Check here for more info. –  Kaster May 2 '13 at 2:24
Ordinary Differential Equation. That's the thing you're trying to solve. –  Kaster May 2 '13 at 2:26

## 1 Answer

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."

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Thanks, I am aware of the pendulum equation for small angles but I was trying to find this for a physics problem involving a lever falling from above the horizontal. However, from the comments I found out that this does not have a closed form in elementary functions, and it is beyond my purposes to get a really messy answer to this question. –  Ovi May 2 '13 at 23:32