Method of Undetermined Coefficients Question?

[XcKyle93]

Homework Statement

Find a suitable form for the general solution of
y'' - 4y' + 4y = 2t² + 4t*e^2t + t*sin(2t)
For respective particular solutions, state where it is a general or a special case. DO NOT evaluate coefficents.

Homework Equations

Y1 = At² + Bt + C
Y2 = (Dt³ + Et²) * e^2t
Y3 = (Ft + G)sin(2t) + (Ht + I)cos(2t)

The Attempt at a Solution

I know the method of undetermined coefficients, and I found the particular solutions. They are right. What does it mean when it asks if a particular solution is of the general or special case? I have no clue.

 

[mighty2000]

In this context, a general solution refers to a solution that can be applied to a wide range of problems, while a special case refers to a solution that is specific to a certain set of conditions or parameters. For example, in the given problem, the particular solution Y1 = At^2 + Bt + C is a general solution because it can be applied to any problem with the given form. On the other hand, the particular solution Y2 = (Dt^3 + Et^2) * e^(2t) is a special case because it is only applicable to problems with a specific form involving t^3 and e^(2t). Similarly, Y3 = (Ft + G)sin(2t) + (Ht + I)cos(2t) is also a special case as it is only applicable to problems involving sine and cosine functions.