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2 years ago

Please help me with this

Mathematics

1 answer:

Novay_Z [31]2 years ago

8 0

430 people in the pool now + p more people ≤ 600 people.

Up to 170 more people could enter the pool.

Equation: 430 + p ≤ 600 (units: "people.")

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Determine the values of the constants r and s such that i(x, y) = x rys is an integrating factor for the given differential equa

garri49 [273]

$\underbrace{y(7xy^2+6)}_{M(x,y)}\,\mathrm dx+\underbrace{x(xy^2-1)}_{N(x,y)}\,\mathrm dy=0$

For the ODE to be exact, we require that $M_y=N_x$ , which we'll verify is not the case here.

$M_y=21xy^2+6$
$N_x=2xy^2-1$

So we distribute an integrating factor $i(x,y)$ across both sides of the ODE to get

$iM\,\mathrm dx+iN\,\mathrm dy=0$

Now for the ODE to be exact, we require $(iM)_y=(iN)_x$ , which in turn means

$i_yM+iM_y=i_xN+iN_x\implies i(M_y-N_x)=i_xN-i_yM$

Suppose $i(x,y)=x^ry^s$ . Then substituting everything into the PDE above, we have

$x^ry^s(19xy^2+7)=rx^{r-1}y^s(x^2y^2-x)-sx^ry^{s-1}(7xy^3+6y)$
$19x^{r+1}y^{s+2}+7x^ry^s=rx^{r+1}y^{s+2}-rx^ry^s-7sx^{r+1}y^{s+2}-6sx^ry^s$
$19x^{r+1}y^{s+2}+7x^ry^s=(r-7s)x^{r+1}y^{s+2}-(r+6s)x^ry^s$
$\implies\begin{cases}r-7s=19\\r+6s=-7\end{cases}\implies r=5,s=-2$

so that our integrating factor is $i(x,y)=x^5y^{-2}$ . Our ODE is now

$(7x^6y+6x^5y^{-1})\,\mathrm dx+(x^7-x^6y^{-2})\,\mathrm dy=0$

Renaming $M(x,y)$ and $N(x,y)$ to our current coefficients, we end up with partial derivatives

$M_y=7x^6-6x^5y^{-2}$
$N_x=7x^6-6x^5y^{-2}$

as desired, so our new ODE is indeed exact.

Next, we're looking for a solution of the form $\Psi(x,y)=C$ . By the chain rule, we have

$\Psi_x=7x^6y+6x^5y^{-1}\implies\Psi=x^7y+x^6y^{-1}+f(y)$

Differentiating with respect to $y$ yields

$\Psi_y=x^7-x^6y^{-2}=x^7-x^6y^{-2}+\dfrac{\mathrm df}{\mathrm dy}$
$\implies\dfrac{\mathrm df}{\mathrm dy}=0\implies f(y)=C$

Thus the solution to the ODE is

$\Psi(x,y)=x^7y+x^6y^{-1}=C$

4 0

3 years ago

Como divido numeros radicales?

worty [1.4K]

Do you have a specific question or like nd question example with numbers

¿Tiene una pregunta específica o como ejemplo de pregunta con números?

4 0

3 years ago

If f(x) = 3x^2 + 1 and g(x) = 1 – x, what is the value of (f – g)(2)?

ziro4ka [17]

F - g(2) = 3x^2 + 1 - ( 1 - x )
= 3x^2 + 1 - 1 + x
= 3x^2 + x
= 3(2)^2 + 2
= 14

the answer is B.14

i am a mathematics teacher. if anything to ask please pm me

4 0

3 years ago

These graphs depict the price and output levels of apples under perfect competition and monopoly market structures. what is the

Ksivusya [100]

The extra amount that Mary would pay for apples under a monopoly instead of perfect competition is B. $3

<h3>Calculations and Parameters:</h3>

From the complete information,

There are images of the prices that can be gotten under a monopoly and the ones that can be gotten in a perfect competition by Mary.

The marginal cost which is graphed against the pound of apples and the price shows Mary would pay $15 and the marginal revenue shows that Mary would pay $12.

Hence, the difference between them, which is the extra amount to be paid would be $15-$12

=$3

Read more about perfect competition here:

brainly.com/question/3914700

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