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ICE Princess25 [194]

3 years ago

14

PLEASE HELP ME WITH THIS WILL GIVE 30 POINTS

1 answer:

maxonik [38]3 years ago

8 0

Answer:

i think the area is 21 srry if im wrong

Step-by-step explanation:

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Consider the following initial-value problem. (x + y)2 dx + (2xy + x2 − 2) dy = 0, y(1) = 1 Let ∂f ∂x = (x + y)2 = x2 + 2xy + y2

IRISSAK [1]

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

Suppose the ODE has a solution of the form $F(x,y)=C$ , with total differential

$\dfrac{\partial F}{\partial x}\,\mathrm dx+\dfrac{\partial F}{\partial y}\,\mathrm dy=0$

This ODE is exact if the mixed partial derivatives are equal, i.e.

$\dfrac{\partial^2F}{\partial y\partial x}=\dfrac{\partial^2F}{\partial x\partial y}$

We have

$\dfrac{\partial F}{\partial x}=(x+y)^2\implies\dfrac{\partial^2F}{\partial y\partial x}=2(x+y)$

$\dfrac{\partial F}{\partial y}=2xy+x^2-2\implies\dfrac{\partial^2F}{\partial x\partial y}=2y+2x=2(x+y)$

so the ODE is indeed exact.

Integrating both sides of

$\dfrac{\partial F}{\partial x}=(x+y)^2$

with respect to $x$ gives

$F(x,y)=\dfrac{(x+y)^3}3+g(y)$

Differentiating both sides with respect to $y$ gives

$\dfrac{\partial F}{\partial y}=2xy+x^2-2=(x+y)^2+\dfrac{\mathrm dg}{\mathrm dy}$

$\implies x^2+2xy-2=x^2+2xy+y^2+\dfrac{\mathrm dg}{\mathrm dy}$

$\implies\dfrac{\mathrm dg}{\mathrm dy}=-y^2-2$

$\implies g(y)=-\dfrac{y^3}3-2y+C$

$\implies F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y+C$

so the general solution to the ODE is

$F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y=C$

Given that $y(1)=1$ , we find

$\dfrac{(1+1)^3}3-\dfrac{1^3}3-2=C\implies C=\dfrac13$

so that the solution to the IVP is

$F(x,y)=\dfrac{(x+y)^3}3-\dfrac{y^3}3-2y=\dfrac13$

$\implies\boxed{(x+y)^3-y^3-6y=1}$

5 0

2 years ago

what is the equation of the line that passes through the points (−1, 3) and (1, 11)? (5 points) 4x − y = 1 4x y = 1 x − 4y = −13

koban [17]

Find slope
(y2-y1)/(x2-x1) is slope
(11-3)/(1-(-1))=8/2=4
y=4x+b
input a point
(1,11)
x=1
y=11
11=4(1)+b
11=4+b
minus 4 from both sides
7=b

the equatin is
y=4x+7

the closest one
minus 4x from both sides
y-4x=7
times everybody by -1
4x-y=-7

last choice is answer

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2 years ago

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r-ruslan [8.4K]

Do you divide 1/2 feet which would be 6 feet? Idk cause I'm only in 7th grade

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3 years ago

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How do you find slope in a X and Y table?

wel

Graph it an watch how it slopes. watch the numbers also. You should b able to tell if you calculate it.

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The length of a blue whale is approximately 3 × 10^4 millimeters; the length of an otter is 2 × 10^3 millimeters. How many times

qaws [65]

The blue whale is 3 x 10^4 / 2 x 10^3 = 15 times longer than the otter.

4 0

3 years ago