# Modelling with exact differential equations?

I'm teaching some very elementary differential equations to engineering students, and their constant question to me is "What's the use of this?" or alternatively "Where would we use this?" Now, I'm not an applied mathematician, but most of the "standard" applications of de's you see in texts seem to include population growth, radioactive decay, cooling, mixing, predator-prey, all of which can be modelled with separable or linear de's.

So what I'm trying to find is a fairly simple example - suitable to first year students with limited mathematics - of a practical "real-world" application which is most easily modelled by an exact differential equation. My web searching has drawn a blank, so I'm open to suggestions!

I was not sure if you were looking for a linear or a nonlinear. If nonlinear, I would have to think about it more.

You can investigate a series resistor, an inductor, and a capacitor (RLC circuit) in an electrical circuit.

This leads to the differential equation:

$$\displaystyle Ri(t) + L \frac{di}{dt} + \frac{1}{C} \int_{-\infty}^{\tau = t} i(\tau)d\tau = v(t)$$

$$\displaystyle \frac{d^2i(t)}{dt^2} +\frac{R}{L}\frac{di(t)}{dt} + \frac{1}{LC}i(t) = 0$$

You can see this circuit and the analysis on the Wiki. It is a second order system and depending on the type of voltage source, leads to intergro-differential or just differential equation. It is easy to analyze and to understand how it works. This can also be used for projects.

You can see the real life circuit, calculations and the Laplace Transform method to solve on the wiki RLC circuit.

Then you can simulate the results using free tools:

• Computable Data Format from Wolfram to simulate the series RLC circuit and see how the DEQ results match real world parameter settings.
• You can also use this slick Circuit Simulator where you build the circuit using point and click and then simulate it.

Lastly, you could actually breadboard the circuit, get a signal generator, a digital multimeter and oscilloscope and measure it in an actual circuit to see how close the ideal world and real world line up (errors abound in nature).

This gives you a very nice approach because with no cost you can review theory, solve a DEQ, run a simulation in two different environments (or use Modelica) and then actually build and test the circuit for no to little cost.

After doing a series circuit, you can switch to a parallel circuit and then a series-parallel circuit.

These components and circuits are used in every electronic device (including appliances, houses, cars, phones, computers...) that we own.

• $+1 \mathbb{TU}$ – Namaste Aug 25 '13 at 0:31
• Thanks - maybe I wasn't clear in my question; I know about de's for LRC circuits and a few other engineering applications (car suspension, beam deflection); what I want is an example of an application which involves solving a non-linear exact de. – Alasdair Aug 25 '13 at 3:18
• Maybe the parachute equation, see: math.stackexchange.com/questions/397461/… – Amzoti Aug 26 '13 at 5:48

Any ODE satisfying a conservation law can be written as an exact equation. So I picked the example of a pendulum, which satisfies a non-linear ODE that is very difficult to solve directly: $$\frac{d^2\theta}{d t^2} + \frac{g}{l}\sin\theta =0$$ But the conservation of energy leads to an exact equation $d(\text{Energy})/dt=0$, which can be integrated to give a first-order ODE that is separable: $$\frac{d\theta}{dt} = \sqrt{{2g\over \ell}\left(\cos\theta-\cos\theta_0\right)}$$

• Could you add more detail and improve your answer? – M. Vinay Jun 9 '14 at 17:04