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3 years ago

Convert 5.45787878... to a fraction. Simplify.

1 answer:

3 0

5.45787878 = 545787878/100000000

divide both numerator and denominator by 2 or any other number

545787878 ÷ 2 = 272893939

100000000 ÷ 2 = 50000000

272893939/50000000 is your answer

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Solve the initial value problems.

slavikrds [6]

Both equations are linear, so I'll use the integrating factor method.

The first ODE

$xy' + (x+1)y = 0 \implies y' + \dfrac{x+1}x y = 0$

has integrating factor

$\exp\left(\displaystyle \int\frac{x+1}x \, dx\right) =\exp\left(x+\ln(x)\right) = xe^x$

In the original equation, multiply both sides by eˣ :

$xe^x y' + (x+1) e^x y = 0$

Observe that

d/dx [xeˣ] = eˣ + xeˣ = (x + 1) eˣ

so that the left side is the derivative of a product, namely

$\left(xe^xy\right)' = 0$

Integrate both sides with respect to x :

$\displaystyle \int \left(xe^xy\right)' \, dx = \int 0 \, dx$

$xe^xy = C$

Solve for y :

$y = \dfrac{C}{xe^x}$

Use the given initial condition to solve for C. When x = 1, y = 2, so

$2 = \dfrac{C}{1\cdot e^1} \implies C = 2e$

Then the particular solution is

$\boxed{y = \dfrac{2e}{xe^x} = \dfrac{2e^{1-x}}x}$

The second ODE

$(1+x^2)y' - 2xy = 0 \implies y' - \dfrac{2x}{1+x^2} y = 0$

has integrating factor

$\exp\left(\displaystyle \int -\frac{2x}{1+x^2} \, dx\right) = \exp\left(-\ln(1+x^2)\right) = \dfrac1{1+x^2}$

Multiply both sides of the equation by 1/(1 + x²) :

$\dfrac1{1+x^2} y' - \dfrac{2x}{(1+x^2)^2} y = 0$

and observe that

d/dx[1/(1 + x²)] = -2x/(1 + x²)²

Then

$\left(\dfrac1{1+x^2}y\right)' = 0$

$\dfrac1{1+x^2}y = C$

$y = C(1 + x^2)$

When x = 0, y = 3, so

$3 = C(1+0^2) \implies C=3$

$\implies \boxed{y = 3(1 + x^2) = 3 + 3x^2}$

7 0

2 years ago

I need help what is 3-2y+(-8y)+7.8

Snowcat [4.5K]

Answer:

10.8-10y OR -10y+ 10.8

Step-by-step explanation:

sO: 3-2y-8y+7.8

3-10y+7.8

10.8-10y / -10y+10.8 Either one

6 0

3 years ago

Read 2 more answers

Andrew walks along a road which can be modeled by the equation y=3x, where (0,0) represents his starting point. When he reaches

Temka [501]

Answer:

(120,0)

Steps:

y-36=-1/3(x-12)

(0)-36=-1/3(x-12)

(-3)(-36)=x-12

108=x-12

120=x

7 0

3 years ago

HELPPPPP

eimsori [14]

I think the answer is b. I’m not positive but should be

8 0

3 years ago

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Solve this problem: –282 – (+1,017) =

gizmo_the_mogwai [7]

This question more or less tries to trick you with the positive number in parentheses, as you would carry out subtraction as normal. However, since the number you are subtracting from is also negative, it would be easier to think about it as adding two positives, and setting aside the negative symbol for the answer.

-282 - 1017 = -(282 + 1017) = -(1299) = -1299

which is your final answer.
-282 - (+1017) = -1299.
Hope that helped! =)

4 0

3 years ago

Read 2 more answers