I am working through the derivation of the WKB approximation and I can't seem to work out how one of the steps is done. I've provided a picture from my lecture notes.
How do they manage to obtain $Y_{1}=-\frac{1}{2}\log{Y_{0}}$ from the previous equation? I have tried working it out and haven't been able to reach the value provided.
They showed $Y_{1}' = - \frac{1}{2} \frac{Y_{0}''}{Y_{0}'}$. Note that $\frac{Y_{0}''}{Y_{0}'} = \left( \log(Y_{0}') \right)'$. Hence $Y_{1}' = -\frac{1}{2} \left(\log(Y_{0}')\right)'$ and the fundamental theorem of calculus gives $Y_{1} = - \frac{1}{2} \log(Y_{0}') + C$ for some $C \in \mathbb{R}$. (It seems, in your attachment, they have the freedom to set $C = 0$.)