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This method is more limited in scope; it applies only to the special case of

where is a constant and has some special form. The advantage of the method is that it does not require any integrations and is therefore quick to use. The homogeneous equation

has the solution

.

To solve the inhomogeneous equation

,

it suffices to find one particular solution . If is any particular solution, then the general solution is

.

The idea behind the method of undetermined coefficients is to look for which is of a form like that of . This is possible only for special functions , but these special cases arise quite frequently in applications.

**Case 1**. We start with the case where is an exponential

.

We look for in a similar form

.

This leads to

.

So the differential equation becomes

.

We can solve this to find

.

This leads to the particular solution

,

and the general solution

.

**Case 2**. If is a polynomial of degree , then is a polynomial of degree . If is a polynomial, we can therefore look for polynomial solutions. Consider

.

The right hand side is a polynomial of degree , so we look for a solution in the same form . This leads to , and

.

To satisfy this, we want to set

.

This leads to , , . So a particular solution is

.

The general solution is

.

We note that the solution

breaks down if , since it would involve a division by zero. More generally, if the equation reads

,

and , with an *n*th degree polynomial, then we can find a particular solution

,

where is some other *n*th degree polynomial as long as . If, on the other hand, , we have to modify the procedure. The modification is simply to include an extra factor in the solution. That is, instead of setting

,

you set

.

Source:** **http://www.math.vt.edu/people/renardym/class_home/firstorder/node2.html

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