What is the constant proportion for y= 2x

Anarel [89]

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—————

Solve the initial value problem:

dy
——— = 2xy², y = 2, when x = – 1.
dx

Separate the variables in the equation above:

$\mathsf{\dfrac{dy}{y^2}=2x\,dx}\\\\
\mathsf{y^{-2}\,dy=2x\,dx}$

Integrate both sides:

$\mathsf{\displaystyle\int\!y^{-2}\,dy=\int\!2x\,dx}\\\\\\
\mathsf{\dfrac{y^{-2+1}}{-2+1}=2\cdot \dfrac{x^{1+1}}{1+1}+C_1}\\\\\\
\mathsf{\dfrac{y^{-1}}{-1}=\diagup\hspace{-7}2\cdot \dfrac{x^2}{\diagup\hspace{-7}2}+C_1}\\\\\\
\mathsf{-\,\dfrac{1}{y}=x^2+C_1}$

$\mathsf{\dfrac{1}{y}=-(x^2+C_1)}$

Take the reciprocal of both sides, and then you have

$\mathsf{y=-\,\dfrac{1}{x^2+C_1}\qquad\qquad where~C_1~is~a~constant\qquad (i)}$

In order to find the value of C₁ , just plug in the equation above those known values for x and y, then solve it for C₁:

y = 2, when x = – 1. So,

$\mathsf{2=-\,\dfrac{1}{1^2+C_1}}\\\\\\
\mathsf{2=-\,\dfrac{1}{1+C_1}}\\\\\\
\mathsf{-\,\dfrac{1}{2}=1+C_1}\\\\\\
\mathsf{-\,\dfrac{1}{2}-1=C_1}\\\\\\
\mathsf{-\,\dfrac{1}{2}-\dfrac{2}{2}=C_1}$

$\mathsf{C_1=-\,\dfrac{3}{2}}$

Substitute that for C₁ into (i), and you have

$\mathsf{y=-\,\dfrac{1}{x^2-\frac{3}{2}}}\\\\\\
\mathsf{y=-\,\dfrac{1}{x^2-\frac{3}{2}}\cdot \dfrac{2}{2}}\\\\\\
\mathsf{y=-\,\dfrac{2}{2x^2-3}}$

So y(– 2) is

$\mathsf{y\big|_{x=-2}=-\,\dfrac{2}{2\cdot (-2)^2-3}}\\\\\\
\mathsf{y\big|_{x=-2}=-\,\dfrac{2}{2\cdot 4-3}}\\\\\\
\mathsf{y\big|_{x=-2}=-\,\dfrac{2}{8-3}}\\\\\\
\mathsf{y\big|_{x=-2}=-\,\dfrac{2}{5}}\quad\longleftarrow\quad\textsf{this is the answer.}$

I hope this helps. =)

Tags: <em>ordinary differential equation ode integration separable variables initial value problem differential integral calculus</em>

7 0

3 years ago

Based on the information provided in the net worth statement, what is the net worth of this person?

BigorU [14]

To find this person's net worth, you will add the assets together to get a total value of what they are worth. Then you will need to subtract the amount of liability (amount owed).

( $487,500 + $265000) - $445500

$307000

This person's net worth is $307000.

5 0

3 years ago

Can someone help me with this?

faltersainse [42]

Answer:

I think it's graph 2 hope that helps

7 0

2 years ago

Read 2 more answers

What is 76/10 in a mixed number??

cestrela7 [59]

A mixed number is in the form x y/z while a improper fraction (76/10) is in the form y/z. We can separate the fraction into two parts. One is 70/10 which = 7 and the other is 6/10 which simplifies to 3/5. So, in order to express 76/10 as a mixed number we must express it as 7 3/5 or 7 6/10. 7 3/5 is equal to 7 6/10.

I hope this helped you!
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5 0

3 years ago

3- A rectangle has an area of 30 square meters arm a perimeter of 34 meters the dimensions , of the rectangle I 9 , a J v ft HA

Leto [7]

If you are looking for the answer to the question above you have to ask yourself what two numbers multiply to 30 and equal 17 when added up. We halve the perimeter to make this easier. So your answer is 15 and 2 because 15 *2=30. If the sides are 15 and 2 then, 15(2) +2(2)=34. There is your answer.

4 0

3 years ago