# What is meant by “method of approximate numerical method” or “method of digital computer” for solving the differential equation of resistive force?

I was reading "motion against resistive forces" in Newtonian Mechanics by A.P. French; here is the excerpt:

[...] In general, the resistive force $\mathbf{R}$ is is a function of speed, so that the statement of Newton's law must be written $$\underset{\text{driving force}}{F_0} - R(v) = m\dfrac{dv}{dt}$$.The resistive force of dry friction is in fact nearly independent of $v$, so that we can put $R(v) \approx \text{constant}$. ...The situation is very different in the case of fluid friction, for which $R(v)$ increases monotonically with $v$ as described $$R(v) = Av + Bv^2$$. ... the net driving force is immediately reduced below $F_0$ as soon as the object has any appreciable velocity it is exposed to a flow of fluid past it at the speed $v$. The statement of Newton's law must now be written $$m\dfrac{dv}{dt} = F_0 - Av - Bv^2$$.

The solving of the above equation is not at all such a simple matter as our familiar problems involving constant forces or forces perpendicular to the velocity. We are now faced with finding the solution to an awkward differential equation. We do not intend to plunge into all the formal mathematics of this problem.Instead, this is a suitable moment to the value of approximate numerical methods - in other words, the method of digital computers, using finitely small intervals.

My question is:

1.What did Mr. French want to mean by the the approximate numerical method or the method of digital computer?

1. What let Mr. French to call the above equation an awkward differential equation?

2. Why can't we solve the equation by normal method?

• I am not sure questions 2 and 3, but for 1, methods like those in this wiki page are used: en.wikipedia.org/wiki/… – tpb261 May 20 '15 at 6:27

## 1 Answer

Your profile lists your age as 17, so I assume you're still at school. At this stage in your physics education you'll only have been exposed to differential equations that have relatively straightforward solutions. I assume the education system does this to avoid putting you off. If you continue studying physics you'll quickly learn that the vast majority of differential equations have no analytical solution and we have to resort to approximate or numerical solutions.

In particular, the equation:

$$m\dfrac{dv}{dt} = F_0 - Av - Bv^2$$

has no simple analytical solution except in some special cases. I assume this is what French means when he says it is awkward, but then he would have to describe the vast majority of differential equations as awkward.

Because so few differential equations have simple solutions, using computational methods to solve them numerically has become a vast industry. Certainly far too broad to be usefully summarised here. I suggest you start with the Wikipedia article on the subject, then Google any specific points that arise from reading that article.