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

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

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

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

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

$\boxed{y = C_1 e^{4x} + C_2 e^{-2x}}$

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

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

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

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

$y = C_1 e^{(-1+i)x} + C_2 e^{(-1-i)x}$

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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Is paper money law a weakness of articles of confederation

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Answer:

What are the weaknesses of the Articles of Confederation?

Weaknesses of the Articles of Confederation

Each state only had one vote in Congress, regardless of size. Congress had not have the power to tax. Congress did not have the power to regulate foreign and interstate commerce. There was no executive branch to enforce any acts passed by Congress.

Explanation:

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