Differential equation is everyone’s cup of tea ;)

Student: I have learnt how to solve differential equations,  but I have no clue if  we ever use them in real life.

Me: ( Very happy with such questions where students try to make sense of the topics that they are learning. )

Yes, differential equations are quite useful in real life and you can see them all around you. Let us try to “see” D.E in a real world scenario.

Change in any variable can be written as a differential. {Will post a visual explanation of differential very soon.}

For example, Velocity = Change in Displacement w.r.t Time
          v = ds/dt

s – displacement, t – time

You can think of ‘d’ as difference, so ds/dt can be:

change in displacement/change in time

Let’s look at one example in detail


Yes, tea!

Once it is served on the table, the temperature of the tea starts changing. (Assuming there isn’t  a hot plate stove on your table 😉

  • Initially, the tea starts cooling rapidly.
  • Slowly, the rate of cooling goes down.
  • Once the tea reaches room temperature, it doesn’t get any cooler. Yes, the rate of cooling is 0 (zero) once the temperature of the tea is same as room temperature.

Let’s say

T – temperature of coffee, R – room temperature

  • Initially, when (T – R) is high, the rate of cooling is also high.
  • As (T – R) decreases, the rate of cooling also goes down.
  • Once T = R, that is  T – R = 0, the rate of cooling is zero.

This implies that the rate of cooling w.r.t time (dT/dt) is directly proportional to the difference of coffee temperature & room temperature (T – R).

           dT/dt = k (T – R)       , k =  proportionality constant

Here is the Linear Differential Equation related to a very simple real world scenario. Upon solving this equation, we will get T(t): a relation between T (temperature of coffee) & t (time).

Let me write it in form of x , y so that it looks more familiar to you

         dy/dx = k (y – R)

Differential Equations can describe how populations change, how tea cools down, how cold drink warms up (once outside the refrigerator), how radioactive material decays and much more.

Hope this article helped  you visualize Differential Equations around you.

My primary job is not just to “teach”, my main focus is to ‘create that desire for learning’.                                                                                                               


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