Complex valued ODE

Question

Say $A(t)$ and $\bar{A}(t)$ are complex valued functions (mutually conjugates). Then if $2A'+A=0$ has solution $\alpha e^{-t/2}$, will $2\bar{A}'+\bar{A}=0$ have solution $\bar{\alpha} e^{-t/2}$, where $\alpha$ and $\bar{\alpha}$ are complex conjugates?

Answer 1

Yes: if $2A'+A=0$, then

$$0=\overline{0}=\overline{2A'(t)+A(t)}=2\overline{A'(t)}+\overline{A(t)}.$$

Since $A(t)=\alpha e^{-t/2}$, we have $\overline{A(t)}= \overline{\alpha}e^{-t/2}$ and $\overline{A'(t)}=-\frac{1}{2}\overline{\alpha}e^{-t/2}$.