# Differential equations and Vector spaces

I was reading Cohn's book on Lie Groups.In introduction part he has given the motivation behind Lie Groups.It is like this

If solution of the differential equation $\frac{dx_{i}}{dt}=u(t)$ is $x_{i}=f_{i}(x,t)$ then we can think of this solution as a linear transformation $x'=S_{t}(x)$ such that each point $x$ can be associated with a new point $x'$ reached after time $t$ .Then family of transformations $(S_{t})$ on the whole space forms a group,with the operation $S_{t}.S_{t'}(x)=S_{t+t'}$.

I could not understand the whole concept of constructing transformations from differential equations and the defined operation

• The group property that you mention fails in general, unless $u=u(x_i)$. Please clarify this point first. – John B Jan 25 '16 at 2:12
• Can you explain this with a simple example of $\frac{dx}{dt}=2t$ in a single variable case ? – Madhu Jan 25 '16 at 11:04