Modeling - First Order vs Higher Order Differential Equations

Question

Throughout my engineering education, I've only witnessed modeling using First Order differential equations. Almost all of my Higher Order knowledge was given either via a spring or heat equation or just randomly generated "Solve this using this" type practice. I never saw real scenarios in which Higher Order differential equations were used.

I would like to understand how engineers and mathematicians can look at a problem and say "we can model this via second order differential equation".

1. Is there a reason why application of First Order DE's are demonstrated more frequently?

2. Can higher order DE's always be used in place of first order? (For example, if two different liquids enter a tank at 2 different rates, can this ever be modeled using higher order DEs?)

Answer 1

The basic rule is that the order of differential equations comes entirely from the relationship used as the basis for modeling. For the stock tank flow examples, the information given is in terms of rates of change, which points to a first-order differential equation, while modeling a spring depends on Newton's second law, which deals with the second derivative of position, so it's a second order differential equation.

If on the other hand suppose you had a tank flow case such as "The rate of flow of a pollutant into a tank is initially 5 kg/min, but changes at a rate equal to $5 Y'_{out}$ where $Y_{out}$ is the flow out of the tank. In this case, you would be modeling the change in the rate, which indicates a second-order differential equation.

There are physical laws that involve higher-order derivatives. For example, in mechanical engineering, beam deformation depends on the 4th derivative of position along the beam. It's harder in general to come up with these higher order relationships, so you don't see them as much in modeling work.