A level revision

Differential equations

Posted in Maths, Core 4

Forming differential equations

A differential equation is simply one which involves the term or similar (i.e. where x and y are any other variable). You can form differential equations when you know or are given information about the rates of change of some quantity (e.g. speed) with respect to another (e.g. time). For example: the rate of change of temperature, T, with respect to time, s, can be written as

When you know more information, typically how a rate of change relates to something else, (e.g. the rate of the decrease in temperature of a cup of tea is proportional to the temperature at that point) you can start to develop your differential equation:

The derivative is negative here because the temperature is decreasing. We can turn this statement into a equation by adding some constant, k:

Here the negatives have been swapped simply to make the equation nicer.

Without any more information this is far as you can go however typically you will be given a set of measurements, e.g. the initial temperature is 90 degrees and decreasing at a rate of 2 degrees a second. With this extra information you can calculate the value of k. This rate of change is -2 degrees a second so:

We've been told the temperature at this point is 90 degrees so:

Hence,

Putting this back into the original equation:

Solving differential equations

To solve a differential equation you need to remove the derivative. Typically this involves integrating the equation:

Example question: Solve

Worked answer: Solving this differential equation is straight forward because the right hand side contains only x. First step is to think of the derivative as a function and then make a simple rearrangement:

We can now integrate both sides:

The left hand of this equation may look a bit strange but is the same as and so integrates to give y. The left hand side is a standard integral, so:

Because this equation contains the constant c it is given the name of 'the general solution' as c can take any value.

Unfortunately not all equations are this simple. Sometimes the equation you may have to solve will contain both x and y on the right hand side. For example:

To handle these types of equations we 'separate the variables' - get all the x terms on one side and the y terms on the other:

Which integrates to:

This is the correct answer but having the natural log on one side isn't all that nice so we should do something about it. By making both sides powers of e we can remove the natural log:

This can be rewritten as:

Because e is a constant and c is a constant e^c is also a constant and is denoted by the letter A, therefore the final answer is:

Particular solutions

As mentioned earlier the equation containing the constant A or c are the general solutions to the differential equation. If you are given some more information such as a set of values of x and y you can find the particular solution of the equation by substituting in the values of x and y into the general solution. For example using the general solution to the equation earlier:

When x = 1 and y = 4:

Hence,

Therefore:

This is the particular solution to the equation.