Method of Undetermined Coefficients

In summary, the method of indetermined coefficients can be applied to find a particular solution of the given equation, as long as it is in the form of a\ddot{y}+b\dot{y}+cy=F(t) where F(t) is the sum or product of polynomials, exponentials, sines, or cosines. Additionally, the equation can be simplified by factoring out a common term, such as 4x.
  • #1
Precursor
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Homework Statement
Decide whether or not the method of indetermined coefficients can be applied to find a particular solution of the given equation:

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The attempt at a solution
I think the answer is yes, because the equation is of the form: [tex]a\ddot{y}+b\dot{y}+cy=F(t)[/tex]where F(t) is the sum or product of polynomials, exponentials, sines, cosines.

Am I correct?
 
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  • #2
Yes it can, you can also factor out the 4x to make it easier since you know what sin2x+cos2 is equal to.
 
  • #3
Oh so I can replace sin2x+cos2x with 1, which gives me a RHS of simply 4x?
 
  • #4
Yep.
 

FAQ: Method of Undetermined Coefficients

What is the method of undetermined coefficients?

The method of undetermined coefficients is a technique used in solving differential equations. It involves assuming a particular form for the solution and finding the coefficients that satisfy the equation.

When is the method of undetermined coefficients used?

This method is typically used when the differential equation has constant coefficients and the non-homogeneous term has a specific form, such as a polynomial, exponential, or trigonometric function.

How does the method of undetermined coefficients work?

The method involves first finding the general solution to the corresponding homogeneous equation. Then, a particular solution is assumed based on the form of the non-homogeneous term. The coefficients in this assumed solution are then solved for and substituted back into the original equation to obtain the particular solution.

What are the limitations of the method of undetermined coefficients?

This method can only be applied to linear differential equations with constant coefficients and specific forms of non-homogeneous terms. It may not work for more complex equations or non-constant coefficients.

How does the method of undetermined coefficients compare to other techniques for solving differential equations?

The method of undetermined coefficients is relatively simple and efficient for specific types of equations. However, for more complex equations, other techniques such as variation of parameters or Laplace transforms may be more effective.

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