Given that $y=y(x)$ and $z=z(x)$ solve for $y$ and $z$ in the following differential equation. \begin{eqnarray} \frac{dy}{dx}=Ae^{-i\alpha x}z,\notag \end{eqnarray} where $A$ is a complex constant and $\alpha$ is a real constant.
The following is my approach: multiplying both sides by $dx$ we get \begin{eqnarray} dy=Ae^{-i\alpha x}zdx\notag \end{eqnarray} Integrating wrt $x$ we obtain the following \begin{eqnarray} y=A\int e^{-i\alpha x}zdx+C,\notag \end{eqnarray} where $C$ is a constant of integration. My question is how to perform the above integration. I need expressions for $y$ and $z$. Kindly assist
$z$ can be an arbitrary continuous function, and $y$ is then an antiderivative of $A e^{-i\alpha x} z(x)$. You can't "perform the integration" unless you have more information, e.g. another differential equation giving you $dz/dx$.