Uniqueness of solution in a scalar equation.

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!

• i found, this question, is the same that math.stackexchange.com/questions/683016/…, but don't have answer. I apreciate any helps. – P3peM4th. Apr 3 '17 at 19:19
• Why not use standard E&U theorem for scalar differential equations? You have linear ODE $y' = a(t) y$ which is smooth with respect to $y$ when $a(t)$ is continuous and continuous with respect to $y$ and $t$. Standard theorem nicely applies here. – Evgeny Apr 4 '17 at 8:19
• Yes!, you are right, today i haved a converse whit a teacher and he sayed something very similar what you propose, thanks! – P3peM4th. Apr 5 '17 at 1:11