Method of Undetermined Coefficients

In summary, the conversation discusses finding a particular solution yp of the given equation y''+y=sinx+xcosx. The individual attempted to solve the problem by first obtaining a temporary solution, then checking for duplicates, multiplying the solution by x, and obtaining the 2nd derivative. However, they ran into a problem where they had more unknowns than equations and couldn't solve the problem. After some time, they were able to figure out their mistake and solve the problem.
  • #1
Chemmjr18
51
1

Homework Statement


Find a particular solution yp of the given equation:
y''+y=sinx+xcosx

Homework Equations

The Attempt at a Solution


1) First I got the temporary yp: Asinx+Bcosx+(Cx+D)sinx+(Ex+F)cosx

2) Then I got the associated homogenous eq. check for to check for duplicates: C1cosx+C2sinx

3) Since there were duplicates, I multiplied the temporary solution by x: Axsinx+Bxcosx+(Cx2+Dx)sinx+(Ex2+Fx)cosx

4) I obtained the 2nd derivative of this and refined it: (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E-D-B)xcosx+(-A-4C-F)xsinx-Cx2cosx-Ex2sinx

5) I then added this to 2) to get (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E)xcosx+(-4C)xsinx=sinx+xcosx

I have more unknowns than I do equations, so I can't solve. Where'd I go wrong? I've been combing through the problem for almost 4 hours...
 
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  • #2
Chemmjr18 said:

Homework Statement


Find a particular solution yp of the given equation:
y''+y=sinx+xcosx

Homework Equations

The Attempt at a Solution


1) First I got the temporary yp: Asinx+Bcosx+(Cx+D)sinx+(Ex+F)cosx

2) Then I got the associated homogenous eq. check for to check for duplicates: C1cosx+C2sinx

3) Since there were duplicates, I multiplied the temporary solution by x: Axsinx+Bxcosx+(Cx2+Dx)sinx+(Ex2+Fx)cosx

4) I obtained the 2nd derivative of this and refined it: (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E-D-B)xcosx+(-A-4C-F)xsinx-Cx2cosx-Ex2sinx

5) I then added this to 2) to get (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E)xcosx+(-4C)xsinx=sinx+xcosx

I have more unknowns than I do equations, so I can't solve. Where'd I go wrong? I've been combing through the problem for almost 4 hours...
Nevermind, I figured it out.
 

1. What is the Method of Undetermined Coefficients?

The Method of Undetermined Coefficients is a technique used in mathematics and physics to solve linear differential equations with constant coefficients. It involves finding a particular solution to the equation by assuming a form for the solution and then determining the unknown coefficients through substitution.

2. When should the Method of Undetermined Coefficients be used?

The Method of Undetermined Coefficients is most effective when the differential equation has a non-homogeneous term that can be written as a sum of polynomial, exponential, trigonometric, and/or logarithmic functions. It is not applicable to equations with variable coefficients or equations with non-constant coefficients.

3. How does the Method of Undetermined Coefficients work?

The Method of Undetermined Coefficients works by assuming a particular solution to the differential equation in the form of the non-homogeneous term. The unknown coefficients are then determined by substituting the assumed solution into the differential equation and solving for the coefficients. The particular solution is then combined with the general solution of the homogeneous equation to obtain the complete solution.

4. What are the limitations of the Method of Undetermined Coefficients?

There are a few limitations to the Method of Undetermined Coefficients. It can only be used for linear differential equations with constant coefficients and non-homogeneous terms that can be written as a sum of certain types of functions. It also does not work for equations with repeated roots or for equations with complex roots.

5. Are there any alternative methods to solve differential equations with non-homogeneous terms?

Yes, there are other methods besides the Method of Undetermined Coefficients that can be used to solve differential equations with non-homogeneous terms. These include the Method of Variation of Parameters, Laplace Transform, and Green's Function method. Each method has its own advantages and limitations, so it is important to choose the most appropriate method for each specific equation.

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