First order equations - method of separation of variables

First order differential equations are those that have a first derivative - y' or dy/dx - as its highest order derivative.

There are two methods to solve first order equations that are taught in MA2051. Deciding which one to use depends on the type of equation. If the equation is linear, you should use the method of integrating factor, which I will cover in another post. Otherwise, you'll need to use separation of variables. Check my post about classifying differential equations if you're not sure what kind of equation you're working with.

Separation of variables is just what it sounds like - your goal is to get all y's and dy's onto one side of the equal sign, and all x's and dx's on the other. Here's a simple example:

dy/dx = y

Don't forget - you can just treat dy/dx like you treat any other fraction. You'll need to multiply both sides by dx and divide both sides by y. The result is:

dy/y = dx

The variables are separated! Now you just integrate both sides.

ln(y) = x + c
e^ln(y) = e^(x+c)
y = e^ce^x
y = Ce^x is your final answer, where big C is equal to the constant e to the power of little c. This is actually an equation that you will use extensively in MA2051, especially in word problems.

Separating your variables can be harder than that example shows. Sometimes it takes some acrobatics. There really isn't any way to outline all these situations in a post. Just make sure you know your algebraic rules thoroughly (see my first post on precalculus topics for some help) and you should always be able to manipulate things into the form you need. Good luck!

Happy calculating,
Rachel