 trasbocohf1

2022-02-15

Throughout my engineering education, Ive Daisie Benitez

Expert

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 $5{Y}_{out}^{\prime }$ where ${Y}_{out}^{\prime }$ 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.

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