Simple Differential Equations

MathsDirect

A differential equation is any equation containing a derivative term. For example

More complicated examples of differential equations, that you will not meet until Further Maths are in the form

A First Order Differential Equation

and

A Second Order Differential Equation

Separation Of Variables

The type of differential equations that you will face in A-level Maths, can be solved using a method called separating the variables. This involves collecting all y terms on one side of the equation, and all the x terms on the other side. You then integrate each side separately.

For Example

Solve the equation

given that y=6 when x=4

To separate the variables, we need to multiply by dx. Since this will leave the dy and dx terms on their own, we must at this point include integral signs

We can now integrate each side separately. Remember that

We therefore get

You must remember to include a constant of integration.

To find the constant, you just need to substitute in the values that were given in the question.

The solution to the differential equation is therefore

Clearly this was a very simple example, but it shows the basic principles for solving simple differential equations. Over the next few pages more complicated examples will be shown.

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