First order differential equations
Differential equation questions in STEP and other advanced mathematics examinations come in several forms. Primarily, they can be first order differential equations, i.e. the highest present derivative is the first, or higher order, i.e. there is a derivative larger than the first. Sometimes, the question will lead you through a particular technique for solving the differential equation. More frequently however, you will need to apply a standard technique to solve, and this article covers the two main techniques for first order differential equations. It is worth noting though, that as with most of STEP and other advanced mathematics examinations you should make sure you know all of the differentiation and integration in A-Level very well; that includes importantly implicit differentiation. With that final point in mind though, lets proceed to our first technique, separation of variables.
Separation of Variables
The first technique, for use on first order 'separable' differential equations, is separation of variables. A first order differential equation
Then, separation of variables means to divide through by
There's really no more to it than that! Though of course the integrals on the left and right hand sides may not be particularly simple ones; they'l often require you to make a substitution. That's one of the reasons why knowing A-Level integration inside out is of key importance; and you should consider going through Module 10 as well for some hints and tips there.
The Method of Integrating Factors
Our second technique exploits the product rule for differentiation to solve first order differential equations that can be written in the form:
Our aim is to be able to write the left hand side as
Now, as a hopefully illuminating example, lets take the simple case:
Then our integrating factor is:
And we have jumping to the final formula above:
So, they're the two techniques you'll need to become familiar with to begin to tackle STEP first order differential equation problems. To really test things out though, lets proceed to work through part of a past STEP question.
Example
Image
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This extract is from Question 6 on STEP III 2008 and it gives a nice introduction to first order differential equations in STEP. Firstly, we're first asked to differentiate
Then, realising the LHS
as required.
So, now we've got ourselves a differential equation for
First then, we need to find our integrating factor. Here it is given by:
Therefore multiplying through and proceeding along the steps we went through in general earlier, or jumping to the final formula, we have:
as required. Then our condition
So, we have
Summary
And that, is that! We've covered the two main techniques for tackling first order differential equations in STEP and other advanced mathematics examinations. Hopefully it should be clear that you need to be happy to differentiate and integrate standard expressions easily, and beyond that it's really just a matter of practice with the key techniques. For STEP III in particular though, higher order differential equations arise. To prepare for them, move on to the next article in this module.