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3 years ago

Which are the solutions of x2 = –13x – 4?

Mathematics

1 answer:

kobusy [5.1K]3 years ago

7 0

x2 = -13x - 4=

2x = -13x - 4=

2x+13x=-13x-4+13x

15x=-4

x=4/15

i hope this is correct

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(x^2y+e^x)dx-x^2dy=0

klio [65]

It looks like the differential equation is

$\left(x^2y + e^x\right) \,\mathrm dx - x^2\,\mathrm dy = 0$

Check for exactness:

$\dfrac{\partial\left(x^2y+e^x\right)}{\partial y} = x^2 \\\\ \dfrac{\partial\left(-x^2\right)}{\partial x} = -2x$

As is, the DE is not exact, so let's try to find an integrating factor µ(x, y) such that

$\mu\left(x^2y + e^x\right) \,\mathrm dx - \mu x^2\,\mathrm dy = 0$

*is* exact. If this modified DE is exact, then

$\dfrac{\partial\left(\mu\left(x^2y+e^x\right)\right)}{\partial y} = \dfrac{\partial\left(-\mu x^2\right)}{\partial x}$

We have

$\dfrac{\partial\left(\mu\left(x^2y+e^x\right)\right)}{\partial y} = \left(x^2y+e^x\right)\dfrac{\partial\mu}{\partial y} + x^2\mu \\\\ \dfrac{\partial\left(-\mu x^2\right)}{\partial x} = -x^2\dfrac{\partial\mu}{\partial x} - 2x\mu \\\\ \implies \left(x^2y+e^x\right)\dfrac{\partial\mu}{\partial y} + x^2\mu = -x^2\dfrac{\partial\mu}{\partial x} - 2x\mu$

Notice that if we let µ(x, y) = µ(x) be independent of y, then ∂µ/∂y = 0 and we can solve for µ :

$x^2\mu = -x^2\dfrac{\mathrm d\mu}{\mathrm dx} - 2x\mu \\\\ (x^2+2x)\mu = -x^2\dfrac{\mathrm d\mu}{\mathrm dx} \\\\ \dfrac{\mathrm d\mu}{\mu} = -\dfrac{x^2+2x}{x^2}\,\mathrm dx \\\\ \dfrac{\mathrm d\mu}{\mu} = \left(-1-\dfrac2x\right)\,\mathrm dx \\\\ \implies \ln|\mu| = -x - 2\ln|x| \\\\ \implies \mu = e^{-x-2\ln|x|} = \dfrac{e^{-x}}{x^2}$

The modified DE,

$\left(e^{-x}y + \dfrac1{x^2}\right) \,\mathrm dx - e^{-x}\,\mathrm dy = 0$

is now exact:

$\dfrac{\partial\left(e^{-x}y+\frac1{x^2}\right)}{\partial y} = e^{-x} \\\\ \dfrac{\partial\left(-e^{-x}\right)}{\partial x} = e^{-x}$

So we look for a solution of the form F(x, y) = C. This solution is such that

$\dfrac{\partial F}{\partial x} = e^{-x}y + \dfrac1{x^2} \\\\ \dfrac{\partial F}{\partial y} = e^{-x}$

Integrate both sides of the first condition with respect to x :

$F(x,y) = -e^{-x}y - \dfrac1x + g(y)$

Differentiate both sides of this with respect to y :

$\dfrac{\partial F}{\partial y} = -e^{-x}+\dfrac{\mathrm dg}{\mathrm dy} = e^{-x} \\\\ \implies \dfrac{\mathrm dg}{\mathrm dy} = 0 \implies g(y) = C$

Then the general solution to the DE is

$F(x,y) = \boxed{-e^{-x}y-\dfrac1x = C}$

5 0

3 years ago

consider the function represented by the equation 16b = 4r - 12. write the equation in function notation, where b is the indepen

katovenus [111]

When you write a function in the form $y=f(x)$ , y is the dependent variable, and x is the independent variable.

So, if b is the independent variable, it means that we have to solve the expression for r.

To do so, we perform the following steps: start with

$16b = 4r - 12$

Add 12 to both sides

$16b+12 = 4r$

Divide both sides by 4:

$4b+3 = r$

So, we wrote the expression in the form $r = f(b) = 4b+3$ , which means that r is the dependent variable and b is the independent variable, as requested.

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3 years ago

Solve 5 / 6 ÷ 1 / 12 ?

Tema [17]

Answer:

Step-by-step explanation:

5/6 ÷ 1/12 ➡ 5/6 × 12/1 = 60/6 ➡ 10

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3 years ago

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A number is written with the following factorization: 22 × 3 × 54 × 8 × 11 2 . Is this factorization a prime factorization? Expl

Flauer [41]

Answer:

This factorization is not a prime factorization.

Step-by-step explanation:

Prime factorization always singles down to prime numbers. The following factorization is not a prime factorization because the factors are not all prime. For example, 22 is not prime.

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Latoya is on her way home in her car. She has driven 15 miles so far, which is three-fourths of the way home. What is the total

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Answer:

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Step-by-step explanation:

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