I don't know how to solve this problem:

Consider the IVP

$$\begin{cases}\dot{x}=f(t,x) \\ x(t_0)=x_0\end{cases}$$

with $\dot{x}=f(t,x)$ and $\ f\in C^1$ and $\Omega\subset\mathbb{R}\times\mathbb{R}^n.$ Let $\phi(t,t_0,x_0)$ be the solution to this IVP.

Prove that $\phi$ is differentiable with respect to $t_0$. Which equation satisfies $\partial\phi_{t_0}$ and under what initial conditions?

The first part is obvious because this IVP satisfies every assumption of the Differentiable dependence on initial conditions theorem. How can we do the second part?

Thanks for your time.

  • $\begingroup$ hint: write the DE as an equivalent IE and recall the Leibniz rule $\endgroup$ – daulomb Jan 30 '18 at 21:30

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