Are $\Re(z)$ and $\Im(z)$ solutions of $z' = az$?

Question

I'm having trouble with a question (I have to answer "true" or "false" and explain it):

We have $a:I \to C,$ a continuous function on $I$, interval of $R$. If the function $z:I \to C$ is a solution of the equation $(E) z'=az$, then $\Re(z)$ and $\Im(z)$ are solutions of $(E)$.

Could you give me some hints? I've been trying to solve that for 1h and can't figure out anything.

Answer 1

Take $a$ to be the constant function $a\colon x\in I \mapsto i \in \mathbb{C}$, and solve the differential equation $z^\prime= i z$. $z\colon x\mapsto e^{ix}$ is solution; can $\operatorname{Re}(z)$ be a solution?