What Is the Appropriate Particular Solution for y''+25y=2xsin(5x)?

In summary, the conversation involves setting up a particular solution for a differential equation using the Method of Undetermined Coefficients. The steps include solving the associated homogeneous equation, separating the variable, and checking the solution with the homogeneous equation. The only error was a mistake in one of the roots, which was corrected to -25 instead of -5. The only question remaining is if the differential equation is y''+25y' or y''+25y, as this would change the particular solution.
  • #1
batmankiller
21
0

Homework Statement


Set up but do not solve for the appropriate particular solution yp for the differential equation
y''+25y=2xsin(5x)
using the Method of Undetermined Coefficients (primes indicate derivatives with respect to x).

Homework Equations


The Attempt at a Solution


I first solved the associated homogeneous equation and got the equation r^2+25r=0 and the roots are 0 and -5. So the equation becomes C1+C2e^(-5x).

I separated 2x as (Ax+B) then seeing as it's being multiplied by sin (5x), I can just do (Ax+B)cos(5x)+(Cx+D)sin(5x). I check this solution with my homogeneous and there are no repeated terms. I tried entering this solution in (Ax+B)cos(5x)+(Cx+D)sin(5x and it keeps telling me I'm wrong. Am I doing something wrong or forgetting a step or two?

Am I doing something horrificly wrong?
 
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  • #2
One of your roots is wrong. It should be -25, not -5. Other than that, it looks good so far.
 
  • #3
Yeah sorry, I was missing the first half of the question, if it says setup but don't solve for the coefficients for just the particular solution, is what I have correct because it keeps saying I'm wrong and I have no idea. Am I missing some steps?
 
  • #4
It looks okay to me. Are you sure the LHS of the differential equation is y''+25y' and not y''+25y? That would change your particular solution a bit.
 
  • #5
oh ****.. yeah it's 25y.. but how does it change my particular solution? o_O
 
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FAQ: What Is the Appropriate Particular Solution for y''+25y=2xsin(5x)?

1. What is the concept of "Undetermined Coefficients"?

"Undetermined coefficients" is a method used in solving differential equations where the solution is assumed to be a combination of known functions multiplied by unknown coefficients. These coefficients are determined by substituting the assumed solution into the original differential equation and solving for the unknown coefficients.

2. When is the "Undetermined Coefficients" method used?

The "Undetermined Coefficients" method is used to solve non-homogeneous linear differential equations with constant coefficients. It is typically used when the non-homogeneous part of the equation consists of known functions such as polynomials, exponential functions, and trigonometric functions.

3. What is the difference between "Undetermined Coefficients" and "Variation of Parameters"?

Both "Undetermined Coefficients" and "Variation of Parameters" are methods used to solve non-homogeneous linear differential equations. However, the main difference is that "Undetermined Coefficients" assumes a specific form for the solution, while "Variation of Parameters" allows for a more general solution by considering a linear combination of known solutions to the corresponding homogeneous equation.

4. Can "Undetermined Coefficients" be used to solve non-linear differential equations?

No, the "Undetermined Coefficients" method can only be used to solve linear differential equations. Non-linear differential equations require different methods such as separation of variables, substitution, or numerical methods for their solution.

5. What are the limitations of the "Undetermined Coefficients" method?

The "Undetermined Coefficients" method may not work if the non-homogeneous part of the equation is not a linear combination of known functions. It also cannot be used for equations with variable coefficients or for equations where the homogeneous solution is a solution to the non-homogeneous equation. In these cases, other methods such as variation of parameters or Laplace transforms may be more suitable for finding the solution.

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