Nonhomogeneous Second Order ODE containing log

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Discussion Overview

The discussion revolves around solving a nonhomogeneous second-order ordinary differential equation (ODE) of the form y'' + 4y' + 4y = e^(-2x)logx. Participants explore different methods for finding a solution, particularly focusing on the method of undetermined coefficients and alternative approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents the ODE and expresses uncertainty about using the method of undetermined coefficients, suggesting a trial solution of the form Ae^(-2x)x^2logx.
  • Another participant suggests considering the operator method as an alternative, indicating that it does not require specific forms for the non-homogeneous function and can provide a solution in integral form.
  • A third participant notes that the method of undetermined coefficients is typically applicable only to specific types of functions on the right-hand side, such as sine, cosine, polynomials, and exponentials.
  • It is mentioned that the method of variation of parameters can be used for any functions, although it may lead to integrals involving non-elementary functions.

Areas of Agreement / Disagreement

Participants express differing opinions on the appropriateness of the method of undetermined coefficients for this particular ODE. There is no consensus on the best approach to take, and multiple methods are proposed without agreement on which is superior.

Contextual Notes

The discussion highlights the limitations of the method of undetermined coefficients and the potential complexity of using variation of parameters, particularly regarding the types of functions involved and the nature of the solutions.

crowy
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Hi guy,

I have this ODE that I'm having problems with

y"+4y'+4y= e^(-2x)logx

Now, Using method of UC to get rid of the RHS I've tried using Ae^(-2x) x^2 logx

However, I'm not quite sure whether that is correct or not as I have never had a question containing logs before
 
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If it is not a must that you have to use method of undetermined coefficients, you can have a look of the operator method. In this case, you don't have to care about what kind of non-homogeneous function you have. At least you can write the solution in integral form. Please refer to my tutorial in http://www.voofie.com/concept/Mathematics/" :

http://www.voofie.com/content/6/introduction-to-differential-equation-and-solving-linear-differential-equations-using-operator-metho/"

For equations with variable coefficients, you can at least try to find the solution using the below method:

http://www.voofie.com/content/84/solving-linear-non-homogeneous-ordinary-differential-equation-with-variable-coefficients-with-operat/"

If you want to see how to solve it step by step, you can try to ask it in http://www.voofie.com/concept/Mathematics/" by submitting a new question, and I am willing to solve it for you.
 
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See the attachement :
 

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In general, the "method of undetermined coefficients" can only be used when the right hand side is the type of function that might be a solution to a homogeneous linear equation with constant coefficients- sine and cosine, polynomials, exponentials, and combinations of those.

The method of "variation of parameters" works with any functions- although it may result in integrals the require non-elementary functions.
 

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