3x^2yz^3-9x^3y^2z^2+12x^2y^3z^2

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

Use the graph of f "(x) below to state x-coordinates of the inflection points for the graph of f(x).

Kipish [7]

0 an 4 are the answers ok

6 0

3 years ago

Read 2 more answers

How do you do this math

Basile [38]

Answer: y = -3x - 6

Step-by-step explanation:

One way to write a <u>linear equation</u> is with slope-intercept form. Slope-intercept form is y = mx + b, where m is the slope, and b is the y-intercept.

Thus, the equation is y = -3x - 6

Hope it helps :) and let me know if you are confused anywhere.

8 0

2 years ago

Read 2 more answers

Each month, 8 magazines arrive at Sydney's home. Write an equation that would show the number of motnths (n) it took for Sydney

Dovator [93]

C.
Since 8 magazines are areiving each month (n) and the total is 56, all you have to do is multiply 8 by n and you get 56, the actual answer would be 7 though.

7 0

2 years ago

hi can someone pls pls plsss! Help with this, I have a test and it’s due on Tuesday so I have time but for now can you pls help

shusha [124]

Answer:

16 units

Step-by-step explanation:

You would take the area of the door (which would be 8 times 3) and subtract the area of the poster (which is 4 times 2).

Next, you would set up your equation: 24-8 which is 16.

Hope this helps! :)

4 0

3 years ago