Click here for introduction to this series and motivation.

Differential Equations

1. An equation is about balancing two sides of a (weight) balance scale.
2. The equations that we study in school (x + 4 = 9) are about finding the number which will balance the scale.
3. But there could be equations for which solution which balances things is a function and not a number!
4. Specifically, you could have equations which relate a change in certain quantity with respect to that quantity.
5. Solving such equations (4), you have to find the function which will balance. In other words, finding functions will obey the rule defined by the equation.
6. Since the solution is a function, it characterizes the evolution of a system. For example, if the equation was about the position of an object and its velocity, the solution will be a function describing the position of the object over time.
7. The actual evolution of the system will depend on where the system started, hence initial conditions are important.
8. Engineering is basically about forecasting (and hence controlling/influencing) systems around us. Things change. So differential equations are at the core of human endeavour: predicting and controlling the world.
9. Since we are studying change, differential equations will show up in every possible field be it social sciences, biology or economics. Unfortunately, most school syllabus on DE will be stuck in Newtonian systems.
10. The exponential function doesn’t show up from thin air. Books get the causality wrong. We don’t have a particular single order differential which is solved by this magical function. Instead, if we hypothesize/model some system in which rate of change of a parameter is proportional to the value of a parameter, then the function which can balance that equation (or evolution of system which obeys this model) has a peculiar shape. We call them exponential functions!

References (Links)

1. This wonderful book on Calculus
2. This Gilbert Strang’s beautiful but hard to Google introduction to exponential function