Solution to nonhomogenous DE using Method of Undetermined Coefficients

In summary, To find the general solution of a given differential equation, use the method of undetermined coefficients and assume a form for the particular solution that includes a polynomial of the same degree as the right hand side and an arbitrary constant. Add this to the solution to the homogeneous equation to get the complete solution.
  • #1
DWill
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Homework Statement


Find the general solution of the given differential equation:

y'' + 9y = (t^2)(e^3t) + 6


Homework Equations





The Attempt at a Solution


I want to first find a particular solution using the method of undetermined coefficients, but I'm not sure what I should "guess" for the form of Y(t). I learned that if g(t) is some exponential function such as e^2t, then I should assume the form looks like Y(t) = Ae^2t, assuming no duplication with solution to the homogenous equation. This doesn't seem to be much harder, but the t^2 and the +6 is confusing me a bit. Any help for how I can get started? Thanks!
 
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  • #2
In general, if the right hand side is a polynomial, you will require a polynomial of the same degree- but since the coefficients are unknown, even powers that do not appear in the right hand side may be required in the solution. Try [tex](At^2+ Bt+ C)e^{3t}[/tex]. For the constant "6" try a constant. You can do that separately using a solution of the form A, and then add them, or do them together with [tex](At^2+ Bt+ C)e^{3t}+ D[/tex]
 

Related to Solution to nonhomogenous DE using Method of Undetermined Coefficients

1. What is the Method of Undetermined Coefficients?

The Method of Undetermined Coefficients is a technique used to solve nonhomogeneous differential equations (DE). It involves finding a particular solution to a nonhomogeneous DE by guessing a form for the solution and then solving for the undetermined coefficients.

2. How is the Method of Undetermined Coefficients different from other methods of solving DEs?

The Method of Undetermined Coefficients is different from other methods because it allows us to find a particular solution without having to solve the entire DE. This saves time and effort, especially for complex DEs.

3. When can the Method of Undetermined Coefficients be used?

The Method of Undetermined Coefficients can be used when the nonhomogeneous term in the DE is in a specific form, such as polynomials, exponential functions, sine and cosine functions, or a combination of these. It is not applicable for all types of nonhomogeneous DEs.

4. What is the general process for using the Method of Undetermined Coefficients?

The general process for using the Method of Undetermined Coefficients is as follows:
1. Identify the form of the nonhomogeneous term in the DE.
2. Guess a particular solution that is in the same form as the nonhomogeneous term, with undetermined coefficients.
3. Substitute the guess into the DE and solve for the undetermined coefficients.
4. Combine the particular solution with the complementary solution to get the general solution.

5. Are there any limitations to using the Method of Undetermined Coefficients?

Yes, there are some limitations to using the Method of Undetermined Coefficients. It can only be used for certain types of nonhomogeneous DEs, and the guess for the particular solution must not be a solution of the complementary solution. Additionally, it may not work for DEs with repeated roots or when the nonhomogeneous term is too complex.

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