What is ] -7.9? in absolute values

Consider the following differential equation. x^2y' + xy = 3 (a) Show that every member of the family of functions y = (3ln(x) +

Veronika [31]

Answer:

Verified

$y(x) = \frac{3Ln(x) + 3}{x}$

$y(x) = \frac{3Ln(x) + 3 - 3Ln(3)}{x}$

Step-by-step explanation:

Question:-

- We are given the following non-homogeneous ODE as follows:

$x^2y' +xy = 3$

- A general solution to the above ODE is also given as:

$y = \frac{3Ln(x) + C }{x}$

- We are to prove that every member of the family of curves defined by the above given function ( y ) is indeed a solution to the given ODE.

Solution:-

- To determine the validity of the solution we will first compute the first derivative of the given function ( y ) as follows. Apply the quotient rule.

$y' = \frac{\frac{d}{dx}( 3Ln(x) + C ) . x - ( 3Ln(x) + C ) . \frac{d}{dx} (x) }{x^2} \\\\y' = \frac{\frac{3}{x}.x - ( 3Ln(x) + C ).(1)}{x^2} \\\\y' = - \frac{3Ln(x) + C - 3}{x^2}$

- Now we will plug in the evaluated first derivative ( y' ) and function ( y ) into the given ODE and prove that right hand side is equal to the left hand side of the equality as follows:

$-\frac{3Ln(x) + C - 3}{x^2}.x^2 + \frac{3Ln(x) + C}{x}.x = 3\\\\-3Ln(x) - C + 3 + 3Ln(x) + C= 3\\\\3 = 3$

- The equality holds true for all values of " C "; hence, the function ( y ) is the general solution to the given ODE.

- To determine the complete solution subjected to the initial conditions y (1) = 3. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:

$y( 1 ) = \frac{3Ln(1) + C }{1} = 3\\\\0 + C = 3, C = 3$

- Therefore, the complete solution to the given ODE can be expressed as:

$y ( x ) = \frac{3Ln(x) + 3 }{x}$

- To determine the complete solution subjected to the initial conditions y (3) = 1. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:

$y(3) = \frac{3Ln(3) + C}{3} = 1\\\\y(3) = 3Ln(3) + C = 3\\\\C = 3 - 3Ln(3)$

- Therefore, the complete solution to the given ODE can be expressed as:

$y(x) = \frac{3Ln(x) + 3 - 3Ln(3)}{y}$

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6 0

3 years ago

Angela is going on vacation to Mexico. She needs to buy new shirts for her trip. She buys 10 new shirts that cost her a total of

vivado [14]

Angela purchased 4 casual shirts and 6 dressy shirts.

$25 × 6 = $150
$18 × 4 = $72

$150 + $72 = $222

hope this helped :)

8 0

3 years ago

The rent for an apartment was $6,600 per year in 2012. If the rent increased at a rate of 4% each year thereafter, use an expone

yan [13]

Answer:

$8,029.91

Step-by-step explanation:

You have to use the exponential growth equation:

A=P(1+r)^t

Where A is the final amount, P is the initial amount, r is the rate of increase, and t is the time in years. So:

A=6,600(1+0.04)^5

(The rate has to be changed into a decimal)

Then, just plug this equation into a calculator (without the A= part, so just 6,600(1+0.04)^5) and you'll get the answer.

8 0

3 years ago

Monica asked her friends to buy a charity raffle ticket for $3. All but 4 of her friends bought a ticket, and she raised $18. Ho

Dafna11 [192]

Answer:

She asked 10 friends to buy tickets.

Step-by-step explanation:

If each ticket sells for 3 dollars, you can get the amount of people that bought the tickets by taking the amount earned divided by the price of the ticket.

$18 / $3 = 6

If 6 friends bought tickets and 4 friends did not, add the two together and you will get how much people Monica asked.

6 + 4 = 10

She asked 10 friends to buy tickets.

5 0

3 years ago

Read 2 more answers

A diver jumps up and over from a cliff in order to avoid rocks and land in the deeper water. The path of the dive can be modeled

lions [1.4K]

Answer:

48.06 to the nearest hundredth.

Step-by-step explanation:

f(x) = -16x^2 + 2x + 48

To find the maximum height we convert to vertex form:

= -16(x^2 + 1/8x) + 48

= -16[x + 1/16)^2 - 1/256] + 48

= -16(x + 1.16)^2 + 16/256 + 48

= 48.0625.

4 0

3 years ago

What is ] -7.9? in absolute values​

What is ] -7.9? in absolute values