First Order DE

First Order Differential Equations

What is a first order differential equation? “Differential’ means it has derivatives or differentials in it. ‘First order’ means the first derivative, dy/dx, for example. (Second order would be d2y/dx2.)

We’ll be learning to solve first order differential equations. That means that if we have an equation dy/dx = 2x, we will solve for a value of y that satisfies this equation. There are two basic steps:

        1. Separate the variables, so that x and y don’t appear in the same term. Also be sure that the dx and dy terms are in the numerator, not the demoninator. Otherwise, we can’t do the next step:

        2. Integrate. Then, if you have enough information, solve for the constant.

For example:

        dy/dx = 2x

        Separate variables:

        dy = 2x dx              Note that we treat “dx” and “dy” as variables.

        Now integrate:

        y = x2 + C

        To solve for the constant, we would need some additional information. Like this: When x = 1, y = 4. So:

        4 = 12 + C          Then  C = 3

        And     y = x2 + 3

Example 1

Example 2

Example 3