Linear Vector Fields
Consider a simple vector fields function $u_t(x)$, which is a simple linear function in $x$, i.e. $u_t(x) = -\theta x$ for $\theta > 0$. Then the function
\[ \psi_t(x_0) = \exp (-\theta t) x_0 \]
defines a flow $\psi$ solving the ODE.
Proof
- Check $\psi_0(x_0) = x_0$
- Check $\frac{d}{dt} \psi_t(x_0) = u_t(\psi_t(x_0))$