Methods to solve $x^2+(x')^2=1$ and similar nonlinear differential equations How do I solve the following:

$x^2 + \left(\frac{dx}{dt}\right)^2 = 1$

Please refer me to an online resource that has worked examples that I can work through to practice.

I have the answer : $ x = \cos{t} $

I need to understand how to get to the answer.
$x^2 + (\frac{dx}{dt})^2 = 1 \Leftrightarrow \frac{dx}{dt} = \sqrt{1-x^2} $ or $\frac{dx}{dt} = -\sqrt{1-x^2} $

Now you have :

$\int \frac{dx}{\sqrt{1-x^2}} = \int dt$ or $ \int -\frac{dx}{\sqrt{1-x^2}} = \int dt $

Which yield you the solutions : $\arcsin(x) = t+c$ or  $-\arcsin(x) = t+c$

Which means : $x(t) = -\sin(t+c_1)$ or $x(t) = \sin(t+c_1)$

Now, depending on your initial conditions, you will be leaded to the answer you mentioned : $x(t) = \cos t$
$$x^2 + (\frac{dx}{dt})^2 = 1$$
$$ \frac{dx}{dt} = \sqrt{1-x^2}$$
$$\frac{dx}{\sqrt{1-x^2}} = dt$$
$$\int \frac{dx}{\sqrt{1-x^2}} = \int dt$$
$$ \arcsin(x)= t + c $$
$$ x(t) = \sin(t+c)$$