If a • b = b, what must the value of a be?

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

The angle of elevation of the top of a tower from a point 42 metres away from it's base on level ground is 26 degrees. find the

Hatshy [7]

Answer:

20.485 Meters

Step-by-step explanation:

So first you wanna draw a diagram. Start with the tower, then on the ground to the left (or right) draw a point. The point will be labeled as 42 m away from the tower. Now draw a line from that point to the top of the tower. This makes your triangle, and that angle you just drew that touches the point is 26 degrees.

Now, since you have a right triangle you can use trig. You know an angle and a side. Specifically, relative to the 26 degree angle you know the adjacent angle and want the opposite, which is the tower. So opposite and adjacent is tangent. So you set up tan(26) = o/42 where o is the opposite side.

So solving you get o = 20.485 meters

3 0

3 years ago

Peyton has collected 120 aluminum cans for recycling. If 20 cans will fit in each blue plastic bag, how many bags will she need

gogolik [260]

Hi Renee!

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We Know:

120 aluminum cans.

20 cans fit in each blue bag.

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Solution:

120 / 20 = 6

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Answer:

She will need 6 bags to carry all the aluminum cans.

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Hope This Helps :)

6 0

2 years ago

The half of the day from midday to midnight can be described as

Serjik [45]

Answer:

24hours divided by 2 is 12

Step-by-step explanation:

8 0

3 years ago

I am very confused lol... So I'm doing a worksheet for Algebra 1 (highschool) and we're doing division and multiplication with f

evablogger [386]

OK first change 3 3/4 to an improper fraction and that will give you 15/4. so you got that part right. Next, change the division sign to multiplication. then flip the 1/2 to 2/1. Then multiply 15 times 2 and that will give you 30. Then multiply 4 times 1 and that will give you 4. then you will have an answer of 30/4. Finally, since you a have a big number over a small number you divide 30 by 4 and that will give you 7 1/2. So it turns out that your little cheat sheet is wrong because when you check your answer you should go on Math-way to check it. Also, a little background about me is that I am in 8th grade and I am taking algebra 1 too.

4 0

3 years ago