Mini question about a complex ode solution

Question

We have $Z(t) \in \mathbb{C}$ and $P=|u_1|^2-|u_2|^2$ with random $u_1(t),u_2(t)\in \mathbb{C}$. If we have the following ode:

\begin{equation} \dot{Z}=i \lambda P Z \end{equation}

is it correct to write the solution of the above as:

\begin{equation} Z=|Z| e^{i\lambda P t} \end{equation}

I am especially asking this because I am wondering if there is a problem putting $P$ in the exponential like that ($P$ should be constant?) in the final solution. Thanks.

Thanks.

Answer 1

The solution for $$\dot{z} = \alpha(t) z$$ is usually written as $$ z(t) = z(0) \cdot \exp{\left ( \int\limits_{0}^{t} \alpha(\tau)\, d\tau \right )}.$$ This solution works for complex-valued functions too — you can just plug it in and check. So I'm willing to say that there is something missing in your formula.