You need to use your solution to part 1 to solve part 2.
You know $\displaystyle \begin{align*} \frac{\mathrm{d}z}{\mathrm{d}x} = \frac{4z}{x} + 2 \end{align*}$. Rewrite it as $\displaystyle \begin{align*} \frac{\mathrm{d}z}{\mathrm{d}x} + \frac{4}{x}\,z = 2 \end{align*}$. This is first order linear, so you can solve for $\displaystyle \begin{align*} z \end{align*}$ using the integrating factor method.
Once you have $\displaystyle \begin{align*} z \end{align*}$, you can get to solution to the original DE because you know $\displaystyle \begin{align*} y = \frac{1}{x} + \frac{1}{z} \end{align*}$.