Question

Find the intergral curves of the equations
dx/yz=dy/zx=dz/xy

Answer 1

Given equation is

$\frac{dx}{yz} = \frac{dy}{xz} = \frac{dz}{yx}$

We can write it as

$xdx = ydy = zdz$

Taking first two

$xdx = ydy$

Integrating both sides

$\frac{y^2}{2} = \frac{x^2}{2} + c_1 \implies y^2 - x^2 = C_1$

Taking last two

$ydy = zdz$

Integrating both sides

$\frac{z^2}{2} = \frac{y^2}{2} + c_2 \implies z^2 - y^2 = C_2$

Thus, $y^2-x^2=C_1$ and $z^2-y^2=C_2$ are integral curves.

Hence solution of the differential equation will be

$\Phi(y^2 - x^2, z^2 -y^2) = 0$

where $\Phi$ is some arbitrary function.