Help with 2 Problems in Differential Equations

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Homework Help Overview

The discussion revolves around two problems in differential equations. The first problem involves finding the general solution to a differential operator equation, while the second problem asks to prove or disprove the existence of constants that satisfy a specific equation involving differential operators.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods for tackling the first problem, including the method of undetermined coefficients and the annihilator method. There is uncertainty regarding the application of these methods. For the second problem, one participant suggests using the FOIL method and mentions the need to apply the product rule when calculating derivatives.

Discussion Status

The discussion is ongoing, with participants exploring different methods and clarifying the problems. There is a recognition of potential solutions involving complex numbers for the second problem, but no consensus has been reached regarding the first problem.

Contextual Notes

One participant noted a typo in the second problem's formulation, which may affect the discussion. There is also mention of the need for clarity in the application of differential operator rules.

ballajr
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Homework Statement



1) Find the General Solution to:
(D3 - D2 + D - I)[y] = t5 + 1

2) Prove or disprove that there are two constants A and B such that:
t2D - tD - 8I = (tD + AI)(tD + BI)



Homework Equations




The Attempt at a Solution



1) I can't figure out how to attempt this one. Doesn't make sense to me.
2) I should FOIL out the RHS of the equation, but I did that on paper and it didn't make too much sense to me.
 
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ballajr said:
1) I can't figure out how to attempt this one. Doesn't make sense to me.

There are two methods that immediately come to mind:

(1)Use the method of Undetermined Coefficients

(2)Use the annihilator method

2) I should FOIL out the RHS of the equation, but I did that on paper and it didn't make too much sense to me.

Show us what you've got, and keep in mind that to calculate something like D(tD) you need to use the product rule.
 
There was a typo in #2:

2) Prove or disprove that there are two constants A and B such that:
t2D2 - tD - 8I = (tD + AI)(tD + BI)
 
There are solutions for A and B if you permit complex solutions.
 

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