Answer without referring back to the text. Fill in the blank. (Enter your answer

Question

Answer without referring back to the text. Fill in the blank. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.)

If y = 3 x + x2 + 8ex

is a solution of a homogeneous fourth-order linear differential equation with constant coefficients, then the roots of the auxiliary equation are?

I don't understand how the answer is supposed to be formated (what are multiplicities?) or how to find the auxilary equation. Any help in producing the correct answer would be greatly appreciated! Thanks in advance.

Explanation / Answer

y = 3 – x +x2 + 8ex

i.e., y = (3 – x +x2 + 8ex)e0x

The equation is of the form

Y = (C1 + C2x + C3x2 )emx + C4 enx

Where, m is repeated root (thrice) and n is the fourth root

On comparing,

C1 = 3 , C2 = -1, C3 = 1 , C4 = 8, m = 0, n = 1

The roots are 0, 0, 0, 1

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