Why does non-dimensionalization work? I know the steps to do the same. But I do not understand why does it work. Why is that non-dimensionalising an ODE gives us a function with much fewer parameters. What's the intuitive sense here behind this approach?
 A: Basically the idea is that numbers with units must necessarily respond in a simple way when we change our system of units. If, say, I change the basic unit of length to be twice as big, then all numbers with $\text{length}^k$ in their units will get divided by $2^k$ accordingly. 
As a result of this, we can choose our system of units to entirely cancel certain parameters from the problem. Essentially the simplest possible example is $y'=ky,y(0)=y_0$. In this case we can introduce characteristic length $y_C$ and characteristic time $t_C$ with $y=y_C x,t=t_C s$. This converts the equation to
$$\frac{y_C}{t_C} \frac{dx}{ds} = k y_C x,x(0)=y_0/y_C.$$
Now again, I can pick $y_C,t_C$ to be whatever I want depending on my purposes, but a natural choice is $y_C=y_0,t_C=1/k$, so that the equation becomes
$$\frac{dx}{ds} = x,x(0)=1.$$
In reality $k$ and $y_0$ are still in the problem but they only reappear when we go back to express the solution of the problem in terms of some other system of units.
