The problem is to solve for the general solution of:

$$yy' + x = (x^2 + y^2)^{1/2}$$

My teacher said to homogenize the equation as a hint during our exam, and afterwards in the solutions he gave this as the homogenized equation before solving it:

$$y' = \frac{\left[1 + \left(\frac yx\right)^2\right]^{1/2} - 1}{\frac yx}$$

I understand that this rearrangement helps us solve the equation by substituting v = (y/x) and going from there, but I still do not understand how this is "homogenizing" the equation. What does that mean exactly, in this context? I thought that a homogeneous equation was an equation with all terms raised to the same power, which is what we now have, I guess. My teacher kept emphasizing that homogeneous equations are where you try to rearrange and get (y/x) in the and substitute for that, but I still don't really understand why. Can anyone kindly explain this example in more depth? Thanks.