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Differential Equations

Differential equations are often referred to as mathematical models. Explain what the phrase
“mathematical model” can mean, experiences with
mathematical models, and how the mathematical use of the word model is similar to and/or
different from the everyday use of the word model (e.g. fashion model, model airplane,
model student, etc.)

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Paul F. | Highly Experienced Physics and STEM Calculus Math TutorHighly Experienced Physics and STEM Calc...
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A model is a replica of something original or real
For example, architects build out of cardboard, small models of a proposed building.
An equation, differential, integral, quadratic or even just linear, models the natural world in some respect.
For example again  y = x2 models a parablola that might be the shape of your satellite antenna.
A differential equation models motion of an object, usually with respect to time.
A pendulum Diff Equation might be  dx/ dt2 = 15
If you solve for x(t), you get the position of the pendulum with time.
So it models the motion precisely.
 
It is said that Mathematics is the Language of Science [the Natural Physical World]
That is because it describes it well and is thus an intellectual model of physical phenomena.