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4 years ago

The square of an integer number

1 answer:

3 0

Answer: In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3 × 3.

Step-by-step explanation:

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Solve the initial value problem where y′′+4y′−21 y=0, y(1)=1, y′(1)=0 . Use t as the independent variable.

igor_vitrenko [27]

Answer:

$y = \frac{7}{10} e^{3(t - 1)} + \frac{3}{10}e^{-7(t - 1)}$

Step-by-step explanation:

y′′ + 4y′ − 21y = 0

The auxiliary equation is given by

m² + 4m - 21 = 0

We solve this using the quadratic formula. So

$m = \frac{-4 +/- \sqrt{4^{2} - 4 X 1 X (-21))} }{2 X 1}\\ = \frac{-4 +/- \sqrt{16 + 84} }{2}\\= \frac{-4 +/- \sqrt{100} }{2}\\= \frac{-4 +/- 10 }{2}\\= -2 +/- 5\\= -2 + 5 or -2 -5\\= 3 or -7$

So, the solution of the equation is

$y = Ae^{m_{1} t} + Be^{m_{2} t}$

where m₁ = 3 and m₂ = -7.

So,

$y = Ae^{3t} + Be^{-7t}$

Also,

$y' = 3Ae^{3t} - 7e^{-7t}$

Since y(1) = 1 and y'(1) = 0, we substitute them into the equations above. So,

$y(1) = Ae^{3X1} + Be^{-7X1}\\1 = Ae^{3} + Be^{-7}\\Ae^{3} + Be^{-7} = 1 (1)$

$y'(1) = 3Ae^{3X1} - 7Be^{-7X1}\\0 = 3Ae^{3} - 7Be^{-7}\\3Ae^{3} - 7Be^{-7} = 0 \\3Ae^{3} = 7Be^{-7}\\A = \frac{7}{3} Be^{-10}$

Substituting A into (1) above, we have

$\frac{7}{3}B e^{-10}e^{3} + Be^{-7} = 1 \\\frac{7}{3}B e^{-7} + Be^{-7} = 1\\\frac{10}{3}B e^{-7} = 1\\B = \frac{3}{10} e^{7}$

Substituting B into A, we have

$A = \frac{7}{3} \frac{3}{10} e^{7}e^{-10}\\A = \frac{7}{10} e^{-3}$

Substituting A and B into y, we have

$y = Ae^{3t} + Be^{-7t}\\y = \frac{7}{10} e^{-3}e^{3t} + \frac{3}{10} e^{7}e^{-7t}\\y = \frac{7}{10} e^{3(t - 1)} + \frac{3}{10}e^{-7(t - 1)}$

So the solution to the differential equation is

$y = \frac{7}{10} e^{3(t - 1)} + \frac{3}{10}e^{-7(t - 1)}$

6 0

4 years ago

Find the slope of the line drawn below

Marysya12 [62]

Answer:

slope is undefined

Step-by-step explanation:

Calculate the slope m using the slope formula

m = $\frac{y_{2}-y_{1} }{x_{2}-x_{1} }$

with (x₁, y₁ ) = (1, 2) and (x₂, y₂ ) = (1, - 3)

m = $\frac{-3-2}{1-1}$ = $\frac{-5}{0}$

Since division by zero is undefined then the slope of the line is undefined

4 0

3 years ago

Select the correct answer.which graph shows a line with a slope of 0?

dusya [7]

Step-by-step explanation:

the first video man and share with you search the Internet resources to help you find the best service providers to get your free and

4 0

2 years ago

The picture below shows Damarian standing at a corner of a field that is in the shape of a rhombus. His friend is standing in th

Nitella [24]

Answer:

The distance between Damarian and his friend is approximately 127 feet

Step-by-step explanation:

The four sides of the rhombus are equal

The diagonals of the rhombus bisects each other and perpendicular to each other

Look to the attached figure

∵ ADCF is a rhombus

∵ The distance of each side of the field is 90 feet

- The four sides of the rhombus are equal

∴ AD = DC = CF = FA = 90

∵ AC and DF are its diagonals

∴ AC ⊥ DF and bisects each other

∵ D is the position of Damarian

∵ F is the position of his friend

∵ The distance between the other two corners of the field is

approximately 128 feet

∴ AC = 128 feet

∵ AC and DF bisects each other at point I

∴ AI = CI

- That means each part is one-half AC

∴ AI = CI = 128 ÷ 2 = 64 feet

Now let us use any right triangle of the four right triangles in the figure to find the length of one-half the other diagonal

In Δ FIC

∵ ∠FIC is a right angle

∵ FC = 90 ⇒ The hypotenuse of the Δ

∵ CI = 64 ⇒ one leg of the Δ

- By using Pythagoras Theorem

∴ (FC²) = (IF)² + (CI)²

- Substitute the values if CI and FC in the formula of Pythagoras

∵ (90)² = (IF)² + (64)²

∴ 8100 = (IF)² + 4096

- Subtract 4096 from both sides

∴ 4004 = (IF)²

- Take √ for both sides

∴ 63.277 = IF

∵ DI = IF = $\frac{1}{2}$ DF

- Multiply DI by 2 to find DF

∴ DF = 126.554

- Round it the the nearest whole number

∴ DF = 127 feet

DF represents the distance between Damarian and his friend

The distance between Damarian and his friend is approximately 127 feet

8 0

3 years ago

Billy wants to buy 657 mugs the mugs are sold for price of 3 for $4 how much was Billy pay for all the mugsb

stira [4]

Answer:

Billy will pay $876 for all the mugs.

Step-by-step explanation:

$\frac{657}{3}=219$ ex]

219

x 4

------

$876

➳ Hope this helps!

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5 0

4 years ago