Help with 2 Problems in Differential Equations

[ballajr]

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.

 

[gabbagabbahey]

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.

 

[ballajr]

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)

 

[Mark44]

There are solutions for A and B if you permit complex solutions.