Show that there is at most one entire function satisfying the following condition. 
Show that there is at most one entire function $f:\mathbb C\to \mathbb C$ with $f(0)=2+3i$ satisfying $$f'(z)=\sin(z)f(z)+e^{z^2}$$ for all $z\in\mathbb C$.

My question: Although this is a problem I encountered in a course for complex analysis I kept wondering if I can use theory for differential equations to solve this: The function $g(f):=\sin(z)f+e^{z^2}$ is continuous in $f$ as a polynomial which means it satisfies a local Lipschitz-condition and thus the corresponding IVP $f'=g(f),\ f(0)=2+3i$ should have a unique solution on $\mathbb C$ (by Picard-Lindelöf). This means there can't be more than one such function. Does this work?

That's right, if you have a proof of Picard-Lindelöf that works for holomorphic functions in the plane.
Luckily you don't need any non-trivial theorems here, because first-order linear equations are trivial via integrating factors. Say $f_1$ and $f_2$ are two solutions, and $h=f_1-f_2$. Then $h'(z)-h(z)\sin(z)=0$, so if $\mu(z)=e^{\cos(z)}$ then $(\mu h)'=0$...
