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Plz use Differential equation method to solve this:

Oduvanchick [21]

Answer:

Step-by-step explanation:

We have the differential equation $y' = y + \frac{x}{y}$ with initial conditions $y(0)=1$ .

First, notice that the equation can be rewritten as

$y'-y =xy^{-1}$ ,

which is a Bernoulli equation. Once we have recognized the type of the equation we know how to continue. Recall that a Bernoulli equation has the general form

$y'+p(x)y=q(x)y^n$ .

In this particular case we have $n=-1$ . This kind of equation is solved by the change of variable $z=y^{1-n}$ . In our exercise we get $z=y^{1-(-1)}=y^2$ . Now we take derivatives and get

$z'=2yy'$ which es equivalent to $\frac{z'}{2y}=y'$ .

Then, we substitute the value of $y'$ we have obtained in the original equation:

$\frac{z'}{2y}-y = xy^{-1}$ .

The next step is to multiply the whole equation by $2y$ , in order to eliminate the denominator of $z'$ . Thus,

$z' -2y^2=2x$ .

Recall that $y^2=z$ , then

$z' -2z=2x$ .

This last equation is a linear equation, which has general solution

$z(x) = \exp\left(-\int(-2)dx\right)\left(\int 2x \exp\left(\int(-2)dx + C\right)\right)$ .

So, let us calculate the integral that appear in the formula:

$\int(-2)dx = -2x$

$\int 2x e^{-2x}dx = -\frac{\left(2x+1\right)e^{-2x}}{2}$ .

Then, the solution for $z$ is

$z(x) = e^{2x}\left(-\frac{\left(2x+1\right)e^{-2x}}{2} + C\right) = -\frac{\left(2x+1\right)}{2} + Ce^{2x}$ .

Now, we return the change of variable:

$(y(x))^2 =-\frac{\left(2x+1\right)}{2} + Ce^{2x}$ .

The last step is to find the value of the constant $C$ . In order to do this, substitute the initial value:

$(y(0))^2 = 1 =\frac{\left(2\cdot 0+1\right)}{2} + Ce^{2\cdot 0} = -\frac{1}{2} + C$ .

Thus, we have the equation

$1=-\frac{1}{2} + C$ that gives $C=\frac{3}{2}$ .

Therefore,

$(y(x))^2 = -\frac{\left(2x+1\right)}{2} + \frac{3}{2}e^{2x}$ .

8 0

4 years ago

A piggy bank contains the same number of quarters, nickels, and dimes. the coins total $4.40. how many of each type of coin does

mestny [16]

Let the number of each coin be n.

The total sum of these coins is $4.40. So we can write:

n Nickles + n Dimes + n Quarters = 4.40

0.05n + 0.10n + 0.25n = 4.40

0.40n = 4.40

n = 11

So this means, there are 11 coins of each type.

5 0

3 years ago

Read 2 more answers

A jacket is on sale for 20% off the regular price. The sale price is $50.00. What is the regular price of the jacket?

Likurg_2 [28]

So it is essentially 80% of the original price , so what i would do is divide it by 8 and then times it by 10 to get the original price , in this case it will be 50/8 (6.25) and then times 6.25 by 10 which will give you the original price of $62.50<span />

7 0

3 years ago

Read 2 more answers

Complete the net worth statement by determining the total assets and total liabilities. What is the net worth?

masha68 [24]

The net worth, based on the net worth statement of total assets and total liabilities, is <u>B) $2,000</u>.

<h3>What is the net worth?</h3>

The net worth of an individual is the difference between the assets owned and the liabilities that the person owes.

<h3>Data and Calculations:</h3><h3>Assets:</h3>

Cash $200

Checking accounts $1,000

Savings account $400

Cars $8,000

Furniture and appliances $1,500

Computers $900

Stereo/Cell phones $400

Clothing $300

Total assets = $12,700

<h3>Liabilities:</h3>

Credit card balances $1,200

Charge accounts balances of $200

Student loans $7,000

Car Loans $2,300

Total liabilities = $10,700

Net worth = $2,000 ($12,700 - $10,700)

<h3>Answer Options:</h3>

A) $200

B) $2,000

C) $10,700

D) $12,700

Thus, the net worth, based on the net worth statement of total assets and total liabilities, is <u>B) $2,000</u>.

Learn more about the net worth of an individual at brainly.com/question/12371230

6 0

3 years ago

Alice make 2 bunches of balloons turre are the same Numbers of red balloons and the same Numbers of blue ballons in each did sir

german

She put 8 balloons in each color group

4 0

4 years ago