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3 years ago

5

What are three pieces of evidence Publius presents to support his claim?

Advanced Placement (AP)

1 answer:

levacccp [35]3 years ago

3 0

Hello. You did not enter the text to which this question refers, which makes it impossible for it to be answered accurately. However, I will try to help you in the best possible way.

It is only possible to know the evidence of a text, after reading it. However, we can consider that evidences are statements that support the arguments developed by the author of the text. These evidences prove or justify giving veracity to what the author said, showing facts, statistical data, examples, among others. In this way, you must look for this type of information, in order to be able to answer your question.

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jek_recluse [69]

Each of these ODEs is linear and homogeneous with constant coefficients, so we only need to find the roots to their respective characteristic equations.

(a) The characteristic equation for

$y'' - 2y' - 8y = 0$

is

$r^2 - 2r - 8 = (r - 4) (r + 2) = 0$

which arises from the ansatz $y = e^{rx}$ .

The characteristic roots are $r=4$ and $r=-2$ . Then the general solution is

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where $C_1,C_2$ are arbitrary constants.

(b) The characteristic equation here is

$25r^2 - 20r + 4 = (5r - 2)^2 = 0$

with a root at $r=\frac25$ of multiplicity 2. Then the general solution is

$\boxed{y = C_1 e^{2/5\,x} + C_2 x e^{2/5\,x}}$

(c) The characteristic equation is

$r^2 + 2r + 2 = (r + 1)^2 + 1 = 0$

with roots at $r = -1 \pm i$ , where $i=\sqrt{-1}$ . Then the general solution is

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Recall Euler's identity,

$e^{ix} = \cos(x) + i \sin(x)$

Then we can rewrite the solution as

$y = C_1 e^{-x} (\cos(x) + i \sin(x)) + C_2 e^{-x} (\cos(x) - i \sin(x))$

or even more simply as

$\boxed{y = C_1 e^{-x} \cos(x) + C_2 e^{-x} \sin(x)}$

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