Solutions to differential equations.

Hi

I have two sets of differential equations

$\displaystyle \Delta_I(x_i, C_I(\lambda))=0$ and $\displaystyle \Delta_{II}(x_i, C_{II}(\lambda))=0$

where $\displaystyle x_i$ are the dependent variables and $\displaystyle C$ are the variable coefficients

which depend on the independant parameter $\displaystyle \lambda$.

Suppose $\displaystyle \Delta_I$ has solution space $\displaystyle X_I$ and

$\displaystyle \Delta_{II}$ has solution space $\displaystyle X_{II}$

My question is what is the Necessary and sufficient conditions for these

two equations to have an non-overlapping solution space (i.e. given $\displaystyle x_i \in X_{I}$ when can we garantee that $\displaystyle x_i \not \in X_{II}~ \forall~ i$)?

Or even how I would start to investigate this?

thanks in advanced

M