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vfiekz [6]
3 years ago
15

Explain how to write 3 X3/8 as the product of a whole number and a unit fraction

Mathematics
1 answer:
mars1129 [50]3 years ago
6 0
A unit fraction means that the fraction has 1 in the numerator. To make 3/8 a unit fraction, you have to make it 1/8 by dividing by 3.

So now you have 3 x 3 x 1/8. You can multiply the 3 and 3.

You are left with 9 x 1/8.
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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
Find the circumference of the circle and round to the nearest tenth
ikadub [295]

Answer:

circumference of circle = 2×pi×r

circumference of circle = 2×3.14×12.7

circumference of circle = 79.756 in

8 0
2 years ago
X=-y+3 <br> 3y+x=5 pleaasseee heellppp meee
love history [14]
3y+(-y+3)=5
2y+3=5
2y=2
y=1
x=-(1)+3
x=2
Your answer is (2,1)
Hope this helps
6 0
2 years ago
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What property is demonstrated by 7+h=h+7
julia-pushkina [17]

Answer:

commutative property

hope that helped, if yes give me brainliest and if no draw my attention by hitting the comment box.

5 0
2 years ago
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The perimeter of a rectangle is 30 in. if its lenth is 3 times the width, find the deminsions​
MAXImum [283]

Step-by-step explanation:

2w+3(2w)=30

8w=30

w=3.75

l=11.25

7 0
3 years ago
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