Algebraic approximation to differential equation

Question

Suppose I have a first order differential equation of the form:

$\frac{dx}{dt} = \frac{1}{\tau}f(x,t)$

where $f(x,t)$ is a nonlinear function. In the limit where the time constant $\tau$ is small, the differential equation can be approximated by an algebraic equation. This is done by rearranging the equation.

$\tau\frac{dx}{dt} = f(x,t)$

ie: as $\tau \rightarrow 0$, $f(x,t) \rightarrow 0$. Physically this makes sense. How can one mathematically justify this though? Without the above manipulation, one ends up with $\frac{dx}{dt} \rightarrow \infty$. Can this somehow be equated to $f(x,t) \rightarrow 0$ ?

Answer 1

For a counterexample, take

$$\dot x=\frac{\sqrt{xt}}\tau.$$

Then

$$2\sqrt x=\frac{2t^{3/2}}{3\tau}+C,\\ x=\left(\frac{t^{3/2}}{3\tau}+C\right)^2$$

As $\tau$ decreases, the solution increases as $\dfrac1{\tau^2}$.

Answer 2

When you consider the equation $\tau \frac{\text{d} x}{\text{d} t} = f(x,t)$ for $\tau \to 0$, the equation is called singularly perturbed. These equations can be analysed using (geometric) singular perturbation theory, or (more generally) multiple time scale analysis. I highly recommend the book

C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015

for more information on the mathematical underpinnings of techniques used to analyse this type of equations.