Solving system $dx = \frac{dy}{2xz} = -\frac {dz}{2xy}$

Question

How to solve the system $$dx = \frac{dy}{2xz} = -\frac {dz}{2xy}?$$ How in general such systems are solved? Thank you in advance.

Answer 1

let us solve $$\frac{dx}{dt} = \frac{1}{4xyz}, \frac{dy}{dt} = \frac{1}{2y}, \frac{dz}{dt} = -\frac{1}{2z}.$$ a solution is $$y = \sqrt{t+B}, z = \sqrt{C - t}$$ and $x$ satisfies the differential equation $$\frac{dx}{dt} = \frac{1}{4x\sqrt{(t+B)(C-t)} }$$ whici\h has a solution $$x = \sqrt{A + \int_0^t\frac{ds}{\sqrt{(s+B)(C-s)}}},\ y = \sqrt{t+B},\ z = \sqrt{C-t} $$ where $A, B, C$ are constants to be determined from initial condition.

Answer 2

What you call "to solve the system" is not clear but anyway, the solutions of this system can be parametrized as $$x^2=a\pm t,\qquad y=b\cos t,\qquad z=b\sin t,$$ for some $(a,b)$ with $b\ne0$.