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2 years ago

If y² = u² + 2ad, then d = ?

Mathematics

1 answer:

GrogVix [38]2 years ago

3 0

Answer:

(y^2-u^2)/2a = d

Step-by-step explanation:

We will move u^2 on left side and divide everything by 2a

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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

Julio has $150. Each week, he saves an additional $10.Write a function ()that models the total amount of money Julio has after w

nadya68 [22]

Answer:

The function that models the total amount of money Julio has after weeks is:

f(x)=150+10x, where:

x is the number of weeks

Step-by-step explanation:

With the information provided, you can say that the total amount of money Julio has would be equal to the initial amount he has plus the result of multiplying the $10 that he saves every week for the number of weeks, which can be expressed as:

f(x)=150+10x, where:

x is the number of weeks

4 0

3 years ago

Read 2 more answers

Choose the equation that best represents the general form of the following expression. x^2/2=7x user: what is the value of "c" i

AlexFokin [52]

X^2/2 = 7x
x^2 = 14x
x^2 - 14x = 0

A quadratic equation is of the form ax^2 + bx + c
Therefore, in 3x^2 + 5x + 7, c = 7

4 0

3 years ago

X^3 = 125 over 27 CAN I PLEASE GET HELP

Zinaida [17]

X³ = 125/27

Cube root both sides to isolate the variable:

∛x³ = ∛(125/27)

x = ∛(125/27)

∛125 = 5, ∛27 = 3

x = 5/3

5 0

3 years ago

I need help with number 3 .

djverab [1.8K]

Answer:

there are 6 activities therefore 360/6 = 60 minutes on each activity

Step-by-step explanation:

5 0

3 years ago

If y² = u² + 2ad, then d = ?​

If y² = u² + 2ad, then d = ?