Purpose of differential equations

Answer 1

Differential equations can be used for many purposes, but ultimately they are simply a way of describing rates of change of variables in an equation relative to each other.

Many real world events can be modeled with differential equations.

For example, imagine that you are observing a cart rolling down a hill, and can measure it's displacement over time as being d = t2 + 3t + 4. Given that, you can calculate it's velocity at any given moment by taking the derivative of the same equation, as velocity is the rate of change of displacement:

d = t2 + 3t + 4

v = dd/dt

∴ v = 2t + 3

Similarly, because acceleration is the rate of change of velocity, you can use the same technique to calculate the rate at which the cart is accelerating:

v = 2t + 3

a = dv/dt

∴ a = 2

This is just one simple example of how differential equations can be used, but the number of applications are endless.