I am starting with describing the streamlines of velocity fields.
Let's work out the following example so as to see where I get stuck:
$$v(x, y, z) = x \hat{i} + y \hat{j} - x \hat{k}$$
Now we set up the differential equation for the field lines:
$$\frac{dx}{x} = \frac{dy}{y} = - \frac{dz}{x}$$
From here on I do not understand why we do what we do; the solution states:
Thus, $z + x = C_1$, $y = C_2 x$. The streamlines are straight halflines emanating from the z-axis and perpendicular to the vector $\hat{i} + \hat{k}$
Please explain why.