Show that $\det \Phi(t)=\det \Phi(t_0) e^{\int_{t_0}^{t} \sum_{j=1}^{n} a_{jj}(s)ds}$ is the unique solution of the scalar equation: $$y'=\big(\sum_{k=1}^{n} a_{kk}(t)\big)y$$ Satisfying the initial condition $y(t_0)=\det \Phi (t_0)$.
I already proved that is a solution. I need only prove uniqueness. I think that i need to apply the existence and uniqueness theorem, i was try to prove suppose there is another solutions, but, i dont obtained nothing. Thanks for any hint or help!