Intro to Differential Equations

This is the first in a series of posts I plan on writing about Engineering Mathematics. It assumes prior knowledge of Calculus.

An equation that contains derivatives of one or more functions is called a differential equation. Consider the equation

\[ y = e^x \]

The derivative of y with respect to x is:

\[ y' = e^x \]

Therefore, combining the equations above, we can say that:

\[ y' = y \]

This is called a differential equation. Now let’s modify our equation:

\[ y = e^{2x} \]

The derivative becomes: \( y' = 2e^{2x} \), so combining both equations, we can say that:

\[ y' = 2y \]

We can further modify our equation: \( y = e^{2x^2} \) so the derivative becomes: \( y' = 4xe^{2x^2} \). We can say that:

\[ y' = 4xy \]

Over the next few lessons we will learn about how to identify and solve differential equations.