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 d^{2}y/dx^{2}.)

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 = x^{2} + C

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

4 = 1^{2} + C Then C = 3

And y = x^{2} + 3