The Method of Undetermined Coefficients

In order to give the complete solution of a nonhomogeneous linear differential equation, Theorem B says that a particular solution must be added to the general solution of the corresponding homogeneous equation.If the nonhomogeneous term d( x) in the general second‐order nonhomogeneous differential equation

  

is of a certain special type, then the method of undetermined coefficientscan be used to obtain a particular solution. The special functions that can be handled by this method are those that have a finite family of derivatives, that is, functions with the property that all their derivatives can be written in terms of just a finite number of other functions.

For example, consider the function d = sin x. Its derivatives are 

 

and the cycle repeats. Notice that all derivatives of d can be written in terms of a finite number of functions. [In this case, they are sin x and cos x, and the set {sin x, cos x} is called the family (of derivatives) of d = sin x.] This is the criterion that describes those nonhomogeneous terms d( x) that make equation (*) susceptible to the method of undetermined coefficients: d must have a finite family.

Here's an example of a function that does not have a finite family of derivatives: d = tan x. Its first four derivatives are

 

Notice that the nth derivative ( n ≥ 1) contains a term involving tan ^n‐1 x, so as higher and higher derivatives are taken, each one will contain a higher and higher power of tan x, so there is no way that all derivatives can be written in terms of a finite number of functions. The method of undetermined coefficients could not be applied if the nonhomogeneous term in (*) were d = tan x. So just what are the functions d( x) whose derivative families are finite? See Table 1.

Example 1: If d( x) = 5 x ², then its family is { x ², x, 1}. Note that any numerical coefficients (such as the 5 in this case) are ignored when determining a function's family.

Example 2: Since the function d( x) = x sin 2 x is the product of x and sin 2 x, the family of d( x) would consist of all products of the family members of the functions x and sin 2 x. That is,

Linear combinations of n functions . A linear combination of two functions y ₁ and y ₂ was defined to be any expression of the form

  

where c ₁ and c ₂ are constants. In general, a lineral, a linear combination of n functions y ₁ y ₂,…, y _n is any expression of the form

  

where c ₁,…, c _n are contants. Using this terminology, the nonhomogeneous terms d( x) which the method of undetermined coefficients is designed to handle are those for which every derivative can be written as a linear combination of the members of a given finite family of functions.

The central idea of the method of undetermined coefficients is this: Form the most general linear combination of the functions in the family of the nonhomogeneous term d( x), substitute this expression into the given nonhomogeneous differential equation, and solve for the coefficients of the linear combination.

Example 3: Find a particular solution of the differential equation

 

As noted in Example 1, the family of d = 5 x ² is { x ², x, 1}; therefore, the most general linear combination of the functions in the family is y = Ax ² + Bx + C (where A, B, and C are the undetermined coefficients). Substituting this into the given differential equation gives

 

Now, combinbing like terms yields

 

In order for this last equation to be an identity, the coefficients of like powers of x on both sides of the equation must be equated. That is, A, B, and C must be chosen so that

 

The first equation immediately gives . Substituting this into the second equation gives , and finally, substituting both of these values into the last equation yields . Therefore, a particular solution of the given differential equation is