Differential equation as a complex function

[wolly]

How did the $$ln(z)$$ appeared in the first image?
Also since this is a diff equation why was y'' replaced with z?
Can someone explain?

 

[blamocur]

The text in the images is all Greek to me, although Google translator says it is Romanian.

Replacing y'' with z is a standard trick in differential equations.
Integrating z'/z gives you log|z|, and integrating 1/.x gives you log|x| respectively.
Where did you find complex functions? Hint: just using 'z' in the formula doesn't necessarily imply complex domain.