y‴−y=x2 has solution ''multiplicity''?

Name: y‴−y=x2 has solution &#039;&#039;multiplicity&#039;&#039;?
Uploaded: 2022-02-23T17:22:57+00:00
Description: y‴−y=x2 has solution &#039;&#039;multiplicity&#039;&#039;?

Question

y‴−y=x2 has solution &#039;&#039;multiplicity&#039;&#039;?

zurilomk4 · Accepted Answer

Your current question has a characteristic polynomial of degree 3. This means that there are 3 roots, when you count multiplicities. In general, let d be the degree of a polynomial

f (x), let r_{1}, r_{2}, \dots, r_{n} Z

be the distinct roots of f(x) with respective multiplicities

m_{1}, m_{2}, \dots, m_{n}

. Then

d = \sum_{m_{i}}

.
Since we find three roots

(1, \frac{1 + i \sqrt{3}}{2}, and \frac{1 - i \sqrt{3}}{2})

, and their multiplicities must add up to 3 (the degree of our characteristic polynomial), the multiplicity of each root must be 1. Thus we do not need more than

k = 0

for each root.
If you're looking for an example of a differential equation that has

m > 1

, you should look at

y^{″} - 2 y^{'} + 1 = 0

. The characteristic polynomial of this equation has the same root repeated twice,

(r - 1) (r - 1)

, meaning that the root

r = 1

has multiplicity 2. Thus the homogeneous solution is

y = C_{1} e^{x} + C_{2} x e^{x}

.

Buck Henry · Accepted Answer

The roots of the characteristic equation

r^{3} - 1 = are r = 1 and r = \frac{- 1 \pm i \sqrt{3}}{2}

, so the general solution of the homogeneous DE

y^{‴} - y = 0

is

y = C_{1} e^{x} + e^{- \frac{x}{2}} (C_{2} \cos (\frac{\sqrt{3}}{2} x) + C_{3} \sin (\frac{\sqrt{3}}{2} x))

.
The usual ansatz for finding a single solution of the non-homogeneous DE works, so I don't see what problems remain?

RizerMix · Accepted Answer

EDIT: What follows is relevant to the question as originally posted. The question was edited after I posted this answer, and this answer is no longer relevant to the revised version of the question. It&#039;s the m=2 case in your notation, so you need xkeαxcos⁡(βx) and xkeαxsin⁡(βx) for k=0..1.

y'''-y=x^{2} has solution ''multiplicity''?

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