# Modeling - First Order vs Higher Order Differential Equations

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?)

• I think this post highlights the problems of terminology in mathematics. "Model Theory", in my line of work, doesn't (normally) deal with modeling "real world" phenomena. Furthermore, the notion of "first order" has a different meaning in Model Theory than in the field of differential equations. Therefore, I have removed the model theory tag. Jul 5, 2016 at 19:37

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.