Solving an Unwieldy Differential Equation

Question

I am at the end of a very long math problem and am left with this equation.

$$\frac{\sqrt{x^{\prime}(t)^2 + y^{\prime}(t)^2}(l \cos \theta)(\sin (2 \theta - T(t)) - \sin (T(t)))-l\sin\theta (\cos (2\theta-T(t))-\cos(T(t))))}{l^2 \sin^2 \theta + l^2 \cos^2 \theta}$$

I know. It's monstrous.

This equation is equal to $\frac{d\theta}{dt}$, and I would like to solve it as a differential equation. The only way I've found that might work is the separation of variables strategy: $$\frac{d\theta}{dt}=f(t)g(\theta)$$ $$\frac{1}{g(\theta)}d\theta=f(t)dt$$ $$\int\frac{1}{g(\theta)}d\theta=\int f(t)dt$$ $$G(\theta)+c_1=F(t)+c_2$$ $$G(\theta)-F(t)=C$$ The issue is that teasing apart the variables in this equation seems like it will take a very, very long time, not to mention taking the integral of what is left. It feels like this is the wrong way to do it.

So, my question is this: how do I solve this differential equation?

Thank you for all the help you can give.

Answer 1

HINT

For starters, simplify the denominator noting that $$ l^2 \sin^2 \theta + l^2 \cos^2 \theta = l^2, $$ and cancel one of the $l$ factor with the numerator. Finally, I would try to simplify the massive trig function on the RHS using various trig summation of sines and cosines to compact it down.