User:Jim.belk/Draft:Method of undetermined coefficients

In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations. In the method, a "guess" is made as to the appropriate form of the solution, and then the values of the coefficients of determined by solving a system of linear equations. The method of undetermined coefficients is closely related to the annihilator method, and can be viewed as a simple case of the method of variation of parameters. A similar method is sometimes used to find solutions to recurrence relations.

Examples

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Example with one coefficient

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Suppose we wish to find a solution to the following linear inhomogeneous differential equation:

 

Because the inhomogeneous part is e3x, we guess (correctly) that the equation has a solution of the form

 

for some constant A. Substituting this guess into the original equation yields:

 

Therefore, one solution to the differential equation above is given by

 

The general solution is a sum of this particular solution with a general solution to the associated homogeneous equation (see the article on linear differential equations).

Example with three coefficients

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Suppose we wish to find a solution to the equation

 

Because the inhomogeneous part is a quadratic polynomial, we guess (correctly) that the equation has a solution of the form

 

for some constants A, B, and C. Substituting this guess into the original equation yields

 

or

 

Setting the coefficients of x2, the coefficients of x, and the constant terms equal gives the following system of linear equations:

 

Solving yields A = 1, B = –3/2, and C = 5/4. Therefore, one solution to the differential equation above is given by

 

Guessing the form

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The first step in the method of undetermined coefficients is to guess the form of the particular solution. This guess is usually based on the inhomogeneous part of the equation:

 

Sometimes the guess listed above does not work, in which case it is necessary to multiply by a power of x. For example, one might guess that the equation

 

has a solution of the form

 

However, this is not correct, as can be seen by substituting this guess into the equation:

 

The correct guess is

 

which yields the solution

 

The annihilator method explains this phenomenon, and can be used to determine the correct guess in a wide variety of situations.

Relation to vector spaces

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In linear algebra, the method of undetermined coefficients can be viewed as a simple application of function spaces and differential operators. Given an equation such as

 

let V be a vector space that contains the inhomogeneous part and which is closed under differentiation:

 

This allows us to write a matrix for the differentiation operator:

 

We can now rewrite the differential equation as a matrix equation:

 
spanned by the functions x3ex, x2ex, xex, and ex.

Examples

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Find a particular solution of the equation

 

The right side t cos t has the form

 

with n=1, α=0, and β=1.

Since α + iβ = i is a simple root of the characteristic equation

 

we should try a particular solution of the form

 
       
       
       

Substituting yp into the differential equation, we have the identity

 

              
                        

              
                        
                        

              

Comparing both sides, we have

                                                  
                                    
                                             
                              

which has the solution   = 0,   = 1/4,   = 1/4,   = 0. We then have a particular solution

 

Consider the following linear inhomogeneous differential equation:

 

This is like the first example above, except that the inhomogeneous part ( ) is not linearly independent to the general solution of the homogeneous part ( ); as a result, we have to multiply our guess by a sufficiently large power of x to make it linearly independent.

Here our guess becomes:

 

By substituting this function and its derivative into the differential equation, one can solve for A:

 
 
 

So, the general solution to this differential equation is thus: