How to Solve Non-Homogeneous ODEs with Multiple Terms?

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    Integral
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Discussion Overview

The discussion centers around solving non-homogeneous ordinary differential equations (ODEs) with multiple terms on the right-hand side. Participants explore methods for finding particular integrals in the context of an upcoming exam, focusing on specific examples of ODEs.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • Ben expresses uncertainty about finding particular integrals for non-homogeneous ODEs with two different terms on the right-hand side.
  • One participant suggests solving the ODEs separately for each term and then adding the results together.
  • Ben acknowledges understanding the second ODE but is unsure about the appropriate guess for the particular integral of the constant term (1).
  • Another participant proposes using polynomial forms (Ax^2 + Bx + C) for the inhomogeneous function of 1 and mentions the operator method as a potential approach.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach for finding particular integrals, as there are differing suggestions regarding methods and forms to use.

Contextual Notes

There are unresolved assumptions regarding the applicability of the operator method and the specific conditions under which different guesses for particular integrals should be made.

Who May Find This Useful

Students preparing for exams in mathematics, particularly those focusing on differential equations and their applications.

benjamince
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Hi guys, this is my first post on the forums - I have a maths exam tomorrow and I'm pretty sure I will need to find the particular integral of a non-homogenous ODE. I find that pretty easy, but I'm not sure how to approach it when there are 2 different terms on the right:

d2y/dt2 - y = 1 + 3cos(2t)

or

(2)d2y/dt2 - dy/dt - y = t/2 + 3e(-t)

Any help would be much appreciated!
Thanks

Ben
 
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why don't you solve

[tex]\ddot{y}-y=1[/tex]

[tex]\ddot{y}-y=3\cos(2t)[/tex]

then add them together
 
Oh ok that makes sense. It might seem silly, but I can do the second one, but I'm not sure what guess to use for the particular integral of 1?
 
You can try powers of x for the inhomogeneous function of 1.
[tex]Ax^2+Bx+C[/tex]

If your exam allows, you may try the operator method. It works for all inhomogeneous function. You may refer to my tutorial in http://www.voofie.com" .

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

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