# applications of linear systems of differential equations

Does anyone know of an application of linear systems of DEs besides multiple spring-mass systems and parallel circuits? I'm looking for an interesting application to show my DE students and we've already spent enough time looking at spring mass systems and circuits. However, these are the only two applications that I could find. Thanks!

• Linear DE are very important in the control theory, such as PID controllers. One other thing that comes to mind might be chemical reactions. Nov 4, 2015 at 16:53

One example is application of Newton's law of cooling to an object submerged in a coolant or heating fluid, which is itself exposed to an ambient environment.

Suppose that a doting father is warming a refrigerated bottle of milk for his infant son by submerging it in a bowl of hot water. If $M$ is the temperature of the milk in the bottle and $W$ is the temperature of the water in the bowl, then

$$\frac{dB}{dT} = k_{WB}(B-W)$$ $$\frac{dW}{dT} = k_{W}(W - T_a) - k_{BW}(W-B)$$

Where the coefficients $K_{WB},$ $K_{BW}$, and $K_{W}$ are given or determined experimentally. This makes for a simple non-homogeneous linear system.

I do inflow/outflow problems with more than one tank. If x(t) represents the amount of salt in a tank as a function of time, and you have brine (or pure water) coming in and thoroughly mixed brine going out, then the differential equation for one tank is

dx/dt = INFLOW RATE - OUTFLOW RATE

If you have two tanks, one with x(t) kg of salt and the other with y(t) kg of salt, and they are interconnected, then you get a system of differential equations:

dx/dt = INFLOW RATE - OUTFLOW RATE dy/dt = INFLOW RATE - OUTFLOW RATE

However the rates can be interdependent.

One popular application is predator prey systems. It's a fairly obvious one, but an interesting one no doubt.

You have the system $$\frac{dx}{dt} = x(a-by)$$ $$\frac{dy}{dt} = cy(x-d)$$ Where $x$ is the population of zebras, $y$ is the population of lions, and $a$,$b$,$c$,$d$ are just constants of your choice. (You can just make $a = b = c = d = 1$. You would need MATLAB, or MAPLE, or any software that can draw solution curves to the system of ODEs. You can then explain to your students what the solution curve means.

There are many states in this predator-prey system.

[A] The population of lions and zebras is relatively small.

[B] The small number of lions allows the zebra population to increase.

[C] The increased number of zebras allows the lion population to increase.

[D] The increase in lion population causes the zebra population to decrease.

[E] The decrease in the zebra population causes the lion population to decrease. We then end back in Stage A.

Personally, I think predator-prey models are a great application of systems of ODEs. Most of the time, students are expected to just solve systems of ODEs, but being able to look at how much information a solution curve to a system of ODEs can tell you is amazing.

• The question was about linear differential equations, no? Nov 5, 2015 at 14:25
• Ah. Woops. Still a nice application, though, if he's interested. Nov 5, 2015 at 20:43
• Thanks Simon. We will actually do predator-prey systems after we're done with linear systems; I was just hoping to show the students a fun linear application before moving on. Nov 6, 2015 at 2:06