Time-invariant IVP How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. When applied to a forced system)?
Thank you.
 A: A system can be treated as  a single ODE for vector-valued function. The comment-answer by Alexei Averchenko explains the concept of time-invariance and some consequences of having this structure:

An ODE $du/dt =f(u,t)$ does not depend on time if $f(u,t_1)=f(u,t_2)$ for any two times $t_1$ and $t_2$. What's great about those systems is that you can consider $f$ a vector field and then it turns out that various kinds of singularities (points where $f$ is zero or not defined) determine the qualitative nature of the system, which can then be studied using topological tools. If there exists a Lagrangian that defines the system in question, you can also define (conserved) energy. 

Concerning the part

What are the implications of a system with this property (esp. When applied to a forced system)?

The implication for the force is that it's constant in time. Some such forces  are conservative (have a potential function associated with it): gravitational or electrostatical. Others are not, e.g., friction. In the former case, there is a conserved "energy"; in the latter you might be able to write down a Liapunov function (a quantity that does not increase with time). 
