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Solving Differential Equations - The Integrating Factor Method

Any equation of the form(1) might be solved using the integrating factor method. This method finds a function ofthat the left hand side can be multiplied by so that the left hand side can be writtenThe integral of this is justso if we can find a function h(x) we can write the solution down as an integral which may (or may not) be evaluated. The integrating factor for (1) isMultiplying (1) bygives

The left hand side can be written

Integrating both sides now gives

Now divide both sides byto give

Example: Solve the differential equation

The integrating factor is

Example: Solve the differential equation

The integrating factor is

Dividing by

If the coefficient ofis not 1 it must be made 1.

Example: Solve the differential equation

Divide byto give

The integrating factor is