Method of undetermined coefficients

In summary, the conversation is discussing the method of undetermined coefficients for solving a differential equation. The person is asking for help with determining the form of the second particular integral. Another person suggests using the auxiliary equation and gives an example. The conversation ends with someone providing a guess for the solution using the least common multiple.
  • #1
s7b
26
0
Hi,

When using the method of undetermined coefficients to solve

2y'' + y' = cos(2x) + 4x + 1

I did it term by term. I figured out the first yp1 for the cos(2x). I'm just not sure what to "guess" as to the form of the yp2 for 4x+1

Does anyone know what to use as the guess?
 
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  • #2
when you solved the auxilary equation if one root was 0, then for a polynomial, the particular integral would be a degree higher. For example if you solved and 0 was a root of the aux. eq'n. and the right side was ax+b, then the PI would be Ax^2+Bx+C
 
  • #3
It is easy to know what to guess.
[D^2+2^2]cos(2x)=0
[D^2](4x+1)=0
Guess the solution to lcm{[2D^2+D],[D^2],[D^2+2^2]}=[D^2][2D+1][D^2+2^2]
 

What is the method of undetermined coefficients?

The method of undetermined coefficients is a technique used to solve non-homogeneous linear differential equations. It involves guessing a particular solution based on the form of the non-homogeneous term and then using this solution to find the coefficients of the general solution.

When is the method of undetermined coefficients used?

The method of undetermined coefficients is used when solving non-homogeneous linear differential equations with constant coefficients. It is most effective when the non-homogeneous term is in the form of a polynomial, exponential, or trigonometric function.

How does the method of undetermined coefficients work?

The method of undetermined coefficients involves two steps: first, guessing a particular solution based on the form of the non-homogeneous term, and second, using this solution to find the coefficients of the general solution. The general solution is then the sum of the particular solution and the complementary solution, which is the solution to the corresponding homogeneous equation.

What is the difference between the method of undetermined coefficients and variation of parameters?

The main difference between the method of undetermined coefficients and variation of parameters is that the former involves guessing a particular solution based on the form of the non-homogeneous term, while the latter involves finding a particular solution by integrating a function that satisfies the homogeneous equation. Variation of parameters is typically used when the non-homogeneous term is not in a simple form, such as when it involves a product of functions.

What are the limitations of the method of undetermined coefficients?

The method of undetermined coefficients can only be used for linear differential equations with constant coefficients and simple forms of non-homogeneous terms. It also relies on guessing the correct form of the particular solution, which may not always be possible. If the guess is incorrect, the method will fail to find a solution.

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