update docstring

Author: ChrisRackauckas

src/systems/diffeqs/basic_transformations.jl

@@ -4,9 +4,38 @@ $(TYPEDSIGNATURES)
 Generates the Liouville transformed set of ODEs, which is the original
 ODE system with a new variable `trJ` appended, corresponding to the
 -tr(Jacobian). This variable is used for properties like uncertainty
-propagation and should be used with a zero initial condition.
+propagation from a given initial distribution density.
 
+For example, if ``u'=p*u`` and `p` follows a probability distribution
+``f(p)``, then the probability density of a future value with a given
+choice of ``p`` is computed by setting the inital `trJ = f(p)`, and
+the final value of `trJ` is the probability of ``u(t)``.
 
+Example:
+
+```julia
+using ModelingToolkit, OrdinaryDiffEq, Test
+
+@parameters t α β γ δ
+@variables x(t) y(t)
+@derivatives D'~t
+
+eqs = [D(x) ~ α*x - β*x*y,
+       D(y) ~ -δ*y + γ*x*y]
+
+sys = ODESystem(eqs)
+sys2 = liouville_transform(sys)
+@variables trJ
+
+u0 = [x => 1.0,
+      y => 1.0,
+      trJ => 1.0]
+
+prob = ODEProblem(sys2,u0,tspan,p)
+sol = solve(prob,Tsit5())
+```
+
+Where `sol[3,:]` is the evolution of `trJ` over time.
 
 Sources:
 
@@ -15,8 +44,6 @@ Optimal Transport Approach
 
 Abhishek Halder, Kooktae Lee, and Raktim Bhattacharya
 https://abhishekhalder.bitbucket.io/F16ACC2013Final.pdf
-
-
 """
 function liouville_transform(sys)
       t = independent_variable(sys)