# Derivative of ODE with respect to parameter

For each value of a parameter $$a \in \mathbb{R}$$, let $$x(a,t)$$ be defined by the ODE

$$\frac{dx}{dt}=F(a,x,t)$$

where $$F$$ is (say) smooth and, for each fixed $$a$$, Lipschitz in $$x$$. It is well known in this case that $$x$$ is well-defined and is smooth with respect to $$a$$.

Question (general): What is known about the derivatives of $$x$$ with respect to $$a$$?

If this question is too general to be helpful, here's a more specific one:

Question (specific): If $$F(a,x,t)$$ is zero unless $$x \in [-B,B]$$ for some constant $$B$$, does this imply that all the derivatives of $$x$$ with respect to $$a$$ are bounded functions of $$t$$?

(An answer to either question could be a reference to a book)

The obvious thing to do is to write down an ODE satisfied by $$y:=\frac{\partial x}{\partial a}$$, which is, I think,

$$\frac{dy}{dt}=\frac{\partial F}{\partial a}+\frac{\partial F}{\partial x}y$$

and this could be used to study the first derivative (and similar for the higher ones). But these equations get a little messy for higher derivatives.