Understanding a reference (Cummins on 2d order ODE) In the first page of The Impulse Response Function and  Ship Motions (Cummins, 1962), it is written that:

We can now write an equation, which has the appearance of a differential equation, relating these various quantities:
$$a(\omega) \ddot x + b(\omega) \dot x + c(\omega) x = F_0\sin(\omega t+ \epsilon)$$
[...]
But suppose $f(t)$ were to be suddenly doubled. Would the instantaneous acceleration be given by
$$ \ddot x = \dfrac{2f(t) - b(\omega)\dot x - c(\omega) x}{a(\omega)} \ ?$$
In general, no!

I don't understand why the above expression does not give the instantaneous acceleration.
 A: In a way, this is the point that the author is making in this paper. The aim seems to be to dissuade people from writing down their model in this way.
The clue is the phrase 

... has the appearance of a differential equation.

In fact, as is made clear in the paragraph leading up to this equation, it is supposed to apply only in the case of a forcing function having an oscillatory form, with a well defined frequency $\omega$. Immediately after the equation, the author re-emphasizes this:

But a differential equation is supposed to relate the instantaneous values of the functions involved.

It's a model (of ship movement, apparently) built on an oversimplification. If you suddenly double the driving force, this appears as a step-function in the time, and the simplifying assumption (that the driving force is just an oscillatory function at a given frequency $\omega$) is violated. 
Of course, if this were a genuine differential equation with frequency-independent coefficients, one would expect the left-hand side (at time $t$) to be proportional to the right-hand side (at time $t$). But the mathematical model has not been constructed from the physical one in such a way as to give such an equation.
The paper goes on to explain what appears to be the Green's function method for tackling (linear) ordinary differential equations (i.e. calculating the response to an impulsive force) and is advocating that people in this field write down the actual differential equation corresponding to the physics, without effective frequency-dependent coefficients.
