8.1 – First Order Linear Differential Equations

In Maths T, you learnt how to solve 2 types of differential equations, namely the separable variable and the homogeneous differential equations. In FMT, you will learn how to solve linear differential equations.

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A differential equation is linear if it is of the form

where a is a function of x. It can be solved by introducing an Integrating Factor, e ^{∫ a dx}.  This term is multiplied to the left and right of the equation, then we will get

integrating both sides, we get

Which is an expression of y in terms of x. This method is very simple, let me give you an example:

Find the general solution of the differential equation

We start by expressing it in the form

Which is

Now that we know the a, we can find the integrating factor,

Note that the integration in the integrating factor doesn’t need a constant, because it will eventually cancel out later. So multiplying it both sides,

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Probably one of the easiest sections, so don’t make mistakes. Notice that the x is not handed over before the integration is done. A common mistake is that you tend to forget to multiply the integrating factor to the right hand side. So be extra careful in calculations. This is the general solution, and the particular solution can be found if more details are given (for example when y = 1, x = 1). Practise. ☺