How to solve this ode, $2 \frac{dA}{dt} (\sqrt{B^{2}-D})-2\frac{dB}{dt}(B)+\frac{dD}{dt}=0$? I was trying to find out the solutions of the following differential equation:
\begin{equation}2 \frac{dA}{dt} (\sqrt{B^{2}-D})-2\frac{dB}{dt}(B)+\frac{dD}{dt}=0\end{equation}
where $A,B,D \in \textit{C}^{\infty}$.
Suggestions?
 A: Well, think about the three-space $\mathbb{R}^3$ with coordinates $p = (A,B,D)\, \in \, \mathbb{R}^3$. Then, if you ignore the parametrization $t \in \mathbb{R}$ for a moment, you realize you have a differential one-form
\begin{equation} \Omega_{p} = 2\, (\sqrt{B^{2}-D}) \, dA - 2\,B \, dB + dD  = 0\end{equation} At each point $p = (A,B,D)\, \in \, \mathbb{R}^3$ the equation $\Omega_p = 0$ defines a two-dimensional tangent plane, which when $p$ moves around the three space gives you a field of tangent planes $\{F_p \, : \, p \in \mathbb{R}^3\}$, also called a two-dimensional distribution. The question is, is there a curve $p = p(t) = (A(t), B(t), D(t))$ which is tangent to the plane field $F_p$.  
In general, the first thing you could check is whether the distribution $F_p$ is integrable (meaning whether it defines a foliation of two dimensional surfaces tangent to the plane field) by checking the conditions of Frobenius' theorem, which in this case means to check whether $ \, \Omega \, \wedge \, d\Omega = 0$. In your case however, the distribution is fairly simple and explicitly integrable. Indeed
$$0 = 2\, (\sqrt{B^{2}-D}) \, dA - 2\,B \, dB + dD = 2\, (\sqrt{B^{2}-D}) \, dA - d(B^2) + dD = $$
$$= 2\, (\sqrt{B^{2}-D}) \, dA - d(B^2 - D) $$
Now divide both sides by $ 2\, (\sqrt{B^{2}-D})$ obtaining the equivalent equation
$$dA \, -\, \frac{d(B^2-D)}{2\, (\sqrt{B^{2}-D})} = 0$$ 
$$dA \, -\, d\Big(\, \sqrt{(B^2-D)}\, \Big) = 0$$ leading to the solution
$$A = \sqrt{(B^2-D)}\, + \, c_0$$ where $c_0$ is a real constant. So if you pick any smooth functions $B=B(t), \, D = D(t)$, that do not violate the restrictions $B(t)^2 \geq D(t)$ of the square root, you get
$$A(t) = \sqrt{\Big(B^2(t)-D(t)\Big)}\, + \, c_0$$ with an arbitrary constant $c_0$.    
