Solve the differential equation ÿ+ 2y + 5y = 4 cos 2t.

The general solution of the differential equation x'(t) = Ax(t), where A is the matrix [0 -1; 1 0], is x(t) = [(c₁² - c₂² + 2ic₁c₂)[tex]e^{(it)[/tex] + (2c₁c₂ - c₁² + c₂²)[tex]e^{(it)[/tex]][1, i].

To solve the differential equation x'(t) = Ax(t), where A is the given matrix [0 -1; 1 0], we can use the method of finding the eigenvalues and eigenvectors.

Step 1: Find the eigenvalues λ of the matrix A by solving the characteristic equation |A - λI| = 0.

The characteristic equation for A is:

|0-λ -1| = 0

|1 0-λ|

Expanding the determinant gives:

(-λ)(-λ) - (-1)(1) = 0

λ² + 1 = 0

Solving the equation, we get two eigenvalues: λ₁ = i and λ₂ = -i.

Step 2: Find the eigenvectors corresponding to each eigenvalue.

For λ₁ = i:

(A - λ₁I)u₁ = 0

|0- i -1| |x₁| = |0|

|1 0- i| |x₂| |0|

Simplifying the equation gives:

-ix₁ - x₂ = 0

x₁ - ix₂ = 0

Solving this system of equations, we get the eigenvector u₁ = [1, i].

For λ₂ = -i:

(A - λ₂I)u₂ = 0

|0+i -1| |x₁| = |0|

|1 0+i| |x₂| |0|

Simplifying the equation gives:

ix₁ - x₂ = 0

x₁ + ix₂ = 0

Solving this system of equations, we get the eigenvector u₂ = [1, -i].

Step 3: Write the general solution as x(t) = c₁x₁(t) + c₂x₂(t), where c₁ and c₂ are constants.

The general solution to the differential equation x'(t) = Ax(t) is:

x(t) = c₁[1, i][tex]e^{(i\lambda_1 t)[/tex] + c₂[1, -i][tex]e^{(i\lambda_2 t)[/tex]

= c₁[1, i][tex]e^{(it)[/tex] + c₂[1, -i][tex]e^{(it)[/tex]

Expanding and simplifying the solution:

x₁(t) = c₁[tex]e^{(it)[/tex] + c₂[tex]e^{(it)[/tex]

x₂(t) = ic₁[tex]e^{(it)[/tex] - ic₂[tex]e^{(it)[/tex]

Therefore, the general solution is:

x(t) = c₁[c₁[tex]e^{(it)[/tex] + c₂[tex]e^{(it)[/tex]][1, i] + c₂[ic₁[tex]e^{(it)[/tex] - ic₂[tex]e^{(it)[/tex]][1, -i]

= (c₁c₁[tex]e^{(it)[/tex] + c₁c₂[tex]e^{(it)[/tex] + ic₁c₁[tex]e^{(it)[/tex] - ic₁c₂[tex]e^{(it)[/tex])[1, i] + (ic₁c₁[tex]e^{(it)[/tex] - ic₁c₂[tex]e^{(it)[/tex] - c₂c₁[tex]e^{(it)[/tex] - c₂c₂[tex]e^{(it)[/tex])[1, -i]

= [(c₁² - c₂² + 2ic₁c₂)[tex]e^{(it)[/tex] + (2c₁c₂ - c₁² + c₂²)[tex]e^{(it)[/tex]][1, i]

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The question is -

What's the general solution (c1x1(t) +c2x2(t)) of a differential equation x'(t) = Ax(t) with a matrix A = [0 -1; 1 0]?

Answer:

A, C, and F

Step-by-step explanation:

Answer:

A

Step-by-step explanation

The condition number for the problem of evaluating the function f(x) = 1/cos(x) near the point x = 1.5708 is approximately 10 raised to power 16

This means that a small change in the input x can cause a large change in the output f(x).

The condition number of a function is a measure of how sensitive the function is to changes in its input. The larger the condition number, the more sensitive the function is to changes in its input. The condition number for the function f(x) = 1/cos(x) can be estimated as follows:

K = |f'(x)| / |f(x)|

where f'(x) is the derivative of f(x) and f(x) is the value of f(x) at the point x. At the point x = 1.5708, f'(x) = -1 and f(x) = 0.017453292519943295. Therefore, the condition number is approximately:

K = |-1| / |0.017453292519943295| = 10 raised to power 16

This means that a small change in the input x can cause a large change in the output f(x). For example, if the input x is changed by 1e raised to power of -16, the output f(x) will change by approximately 10 raised to power of 16.

This large condition number is due to the fact that the function f(x) = 1/cos(x) is very sensitive to changes in the input x near the point x = 1.5708. This is because the cosine function is very close to zero at this point.

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Answer:

[tex](x,y) => (2.5x,2.5y)[/tex]

Step-by-step explanation:

Given

See attachment for ABCD and A'B'C'D'

Required

Determine the dilation rule

Using AB and A'B' as points of references;

[tex]AB = (-1,2)\ to\ (2,1)[/tex]

[tex]A'B' = (-2.5,5)\ to\ (5,2.5)[/tex]

The dilation factor (k) is calculated as:

[tex]k = \frac{A'B'}{AB}[/tex]

This gives:

[tex]k = \frac{(-2.5,5)}{(-1,2)} \ or\ k = \frac{(5,2.5)}{(2,1)}[/tex]

Factorize

[tex]k = \frac{2.5*(-1,2)}{(-1,2)} \ or\ k = \frac{2.5*(2,1)}{(2,1)}[/tex]

[tex]k = 2.5 \ or\ k = 2.5[/tex]

Hence, the scale factor is 2.5

The dilation rule is:

[tex](x,y) => k(x,y)[/tex]

[tex](x,y) => 2.5(x,y)[/tex]

[tex](x,y) => (2.5x,2.5y)[/tex]

Answer:

1.2

Hope it helps :)

[tex]\frac{24}{25}*\frac{5}{4}\\\\\\=\frac{6}{5}[/tex]

Answer:

its c ^u^ and the next to it is b :D

Step-by-step explanation:

edg 21

Answer:

It’s c and the next one is b

Step-by-step explanation:

Just took edge

Answer:

What is the lenght and the other things?

Step-by-step explanation:

Brainliest?

Answer:

see photo

Step-by-step explanation:

Answer:

no

Step-by-step explanation:

the first two numbers is not greater than 13 so no

Answer:

P value = 0.23527 which is greater than 0.10

Step-by-step explanation:

Given :

Phat = proportion of rotten apple = 2/50 = 0.04

1 - phat = 0.96

p = 0.06

Tstatistic = (Phat - p) / sqrt[(Phat(1-Phat))/n]

Tstatistic = (0.04 - 0.06)/sqrt[(0.04(0.96))/50]

Tstatistic = (-0.02) / 0.0277128

Tstatistic = - 0.7216

Pvalue of Tstatistic

P(Z < - 0.7216) = 0.23527 (Pvalue calculator)

Answer:

A

Step-by-step explanation:

Answer: I would say it's B

Step-by-step explanation: The reason is because whiskers are primary to have, Sharp claws is a tool for animals to get food, and large beaks are used to pick at the food so they don't eat the wrong things.

You need 5/2 times the amounts given in the recipe, so you need:

Here we have the recipe for 16 crepes:

You want to make 5 crepes for person, for 8 persons, so you need:

8*5 = 40 crepes.

So you need to make the recipe N times:

16*N = 40

N = 40/16 = 10/4 = 5/2

So you need a fraction of 5/2 times each of the amounts in the recipes.

You will need:

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Answer:

7238 units^3

Step-by-step explanation:

The formula for the volume of a sphere of radius 12 units is

V = (4/3)(pi)r^3.  

Here, with r = 12 units,

V = (4/3)(pi)(12 units)^3 = 7238 units^3

Answer:

V = 2304π units³

Step-by-step explanation:

V = (4/3)πr^3

V = (4/3)π(12)^3

V = (4/3)π(1728)

V = 2304π units³

Answer:

The equation to solve to find the number of yellow marbles =

3y + y = 500

4y = 500

The number of yellow marbles = 125

Step-by-step explanation:

Let the number of

Blue marbles = x

Yellow marbles = y

A large jar contains 500 marbles.

x + y = 500

All of the marbles in the jar are either blue or yellow, and there are 3 times as many blue marbles as yellow marbles.

3y = x

Hence, we substitute 3y for ymx in the above equation

3y + y = 500

4y = 500

y = 500/4

y = 125

The number of yellow marbles = 125

Answer:

C:  y = 5x + 3

Step-by-step explanation:

Note that if x changes from 1 to 3 (the "run"), y changes from 8 to 14 (the "rise").  Thus the slope of the line represented by the table is m = rise/run = 6/2, or m = 3.  Answer choice D is the only one that displays a slope of 3.

Answer:

Answer is D

Step-by-step explanation:

A. 14=8(3), 14≠24

B. 14=3+7 , 14≠10

C. 14=5(3)+3, 14≠18

D.14=3(3)+5, 14=9+5 14=14

Answer:

b is 10 degrees and a is 64 degrees.

Step-by-step explanation:

Properties of a parallelogram needed for this problem: Opposite angles are congruent and adjacent angles are supplementary, meaning they add to 180 degrees.

This means that in the picture, 2a and 12b+8 are equal, while 2a and 5b+2 are supplementary.

From this we can make two equations:

2a=12b+8

and

2a+5b+2=180

If we subtract 5b+2 in the second equation, both equations will equal 2a. I'll explain why that's good. We end up with:

2a=12b+8

and

2a=180-5b-2

Now, we can set the two equations equal to each other because they both equal the same thing. Then simplify.

12b+8=180-5b-2

17b=170

b=10

Knowing b is 10 degrees, we can easily plug that back into one of our original two equations. I'll choose 2a=12b+8.

2a=12(10)+8

2a=120+8

2a=128

a=64

b is 10 degrees and a is 64 degrees.

Answer:

516 miles

Step-by-step explanation:

Answer:

its option B : 18x3 – 12x2 + 9x – 6

Step-by-step explanation:

[tex]12( \frac{ -2 }{3}) {}^{2} +23( \frac{ - 2}{3} )+10= \frac{16}{3} − \frac{46}{3} +10=0[/tex]

12x² + 15x + 8x + 10 has a binomial factor of (3x+2).

ax²+bx+c=0, with a not equal to 0 is a quadratic equation, which is a second-order polynomial equation in a single variable. It has at least one solution because it is a second-order polynomial equation, which is guaranteed by the algebraic fundamental theorem. The answer could be simple or complicated.

We may utilize the factor theorem, which asserts that if a polynomial f(x) has a component (x - a), then f(a) = 0, to identify which polynomial has (3x + 2) as a binomial factor.

In this situation, finding a value of x that reduces 3x + 2 to zero is necessary since we are trying to discover a polynomial that contains (3x + 2) as a component. We obtain x = -2/3 by solving 3x + 2 = 0.

Let's try the polynomials one by one:

(a) 6x³ + 3x² + 4x + 2:

f(-2/3) = 6(-2/3)³ + 3(-2/3)² + 4(-2/3) + 2

= -8/3 - 4/9 - 8/3 + 2

= -32/9,

so (3x + 2) is not a factor.

(b) 12x² + 15x + 8x + 10:

f(-2/3) = 12(-2/3)₂ + 15(-2/3) + 8(-2/3) + 10

= 16/3 - 10 - 16/3 + 10

= 0,

so (3x + 2) is a factor.

(c) 18x³ - 12x² + 9x - 6:

f(-2/3) = 18(-2/3)³ - 12(-2/3)² + 9(-2/3) - 6

= -32/3 + 8/3 - 6 - 6

= -32/3,

so (3x + 2) is not a factor.

(d) 21x⁴ + 7x³ + 6x + 2:

f(-2/3) = 21(-2/3)⁴ + 7(-2/3)³ + 6(-2/3) + 2

= 64/27 - 56/27 - 4 + 2

= 2/27,

so (3x + 2) is not a factor.

Therefore, the only polynomial that has (3x + 2) as a binomial factor is 12x² + 15x + 8x + 10.

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The finite abelian group Z36 x Z48 x Z60 can be expressed as Z2^6 x Z3^4 x Z5. The finite abelian group Z8 x Z9 x Z11 x Z12 can be expressed as Z2^3 x Z3^2 x Z11 x Z2^2 x Z3.

(a) The finite abelian group Z36 x Z48 x Z60 can be expressed as the direct product of its prime power factorizations. The prime factorization of 36 is 2^2 * 3^2, 48 is 2^4 * 3, and 60 is 2^2 * 3 * 5. Therefore, the group can be written as Z2^6 x Z3^4 x Z5.

(b) The finite abelian group Z8 x Z9 x Z11 x Z12 can be expressed as the direct product of its prime power factorizations. The prime factorization of 8 is 2^3, 9 is 3^2, 11 is a prime number, and 12 is 2^2 * 3. Therefore, the group can be written as Z2^3 x Z3^2 x Z11 x Z2^2 x Z3.

In both cases, the group is expressed as the direct product of cyclic groups of prime power orders. The order of the group is the product of the orders of the individual cyclic groups, and the structure of the group is determined by the prime factors and their powers.

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The spring constant, k = 27 N/m. The system is critically damped. The position function of the object is x(t) = e-6t(cos √3t + √3/3 sin √3t).

(a) Given, m = 1 kg, c = 12 and the spring is stretched 0.5 meters by a force of 13.5 N. We need to find the spring constant k. Let's consider the force applied on the spring, F = -kx

Here, x = 0.5 m and F = 13.5 N. So, 13.5 = -k (0.5) => k = -27 N/m

Since, we cannot have negative spring constant. Therefore, k = 27 N/m.

(b) The given differential equation is x" + 12x' + 27x = 0.

On comparing with the general equation of a damped mass-spring system, we get m = 1 kg, c = 12, k = 27.

The damping factor is defined as β = c/2m = 6.

Using the damping factor we can determine the type of the system. c2 - 4mk > 0 - overdamped systemc2 - 4mk = 0 - critically damped systemc2 - 4mk < 0 - underdamped system In this case, c2 - 4mk = 12^2 - 4(1)(27) = 0

This indicates that the system is critically damped.

(c) The general equation for a damped mass-spring system is x(t) = e-βt(A cos ωdt + B sin ωdt)

Where, A and B are constants that are determined using initial conditions. x(0) = A = 1 and x'(0) = -βA + ωdB/dt = -2 => B = (1/ω) (-βA + 2)

Now, we need to determine ω.ω2 = k/m - β2/4m2 = 27/1 - (12/2)2/4(1)(27) = 3ω = √3

Substitute the value of A, B and ω in the general equation, we get x(t) = e-6t(cos √3t + √3/3 sin √3t)

Hence, the position function of the object is x(t) = e-6t(cos √3t + √3/3 sin √3t).

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the coordinates of the reflected point are (4, 2).

When a point is reflected about the y-axis, the x-coordinate is negated while the y-coordinate remains the same.

The original point is (-4, 2). If we reflect this point about the y-axis, the x-coordinate becomes positive, and the y-coordinate remains unchanged.

Negating the x-coordinate, we get:

x-coordinate: -(-4) = 4

y-coordinate: 2 (unchanged)

Given the point (-4, 2), reflecting it about the y-axis would result in the x-coordinate changing its sign while the y-coordinate remains unchanged.

Therefore, the coordinates of the reflected point are (4, 2).

Among the options provided, the correct answer is (c) (4, 2).

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Answer:

5) No

The actual distance between the drone and the fossil is [tex]32\dfrac{3}{4}[/tex] ft

Step-by-step explanation:

3) A. -2/9

(2/3 - 5/6)/3/4 = (-1/6)/(3/4) = (-1/6) × (4/3) = -2/9

4) d. [tex]0.\overline {90}[/tex]

10/11 = 0.9090909...

∴ 10/11 = [tex]0.\overline {90}[/tex]

5) The height of the drone is flying above the ground = [tex]25\dfrac{1}{2}[/tex] ft.

The location of the fossil below the ground = [tex]7\dfrac{1}{4}[/tex] ft.

Taking the ground level as the origin on a number line, we have;

The location of the drone relative to the ground =  + [tex]25\dfrac{1}{2}[/tex] ft. (The positive sign for a point above the origin)

The location of the fossil relative to the ground = [tex]-7\dfrac{1}{4}[/tex] ft. (The negative sign for a point below the origin)

Therefore;

The distance between the drone and the fossil = The difference between the location of the drone and the fossil

The distance between the drone and the fossil = [tex]25\dfrac{1}{2} - \left(-7\dfrac{1}{4} \right)[/tex]

[tex]25\dfrac{1}{2} - \left(-7\dfrac{1}{4} \right) = 25\dfrac{1}{2} + 7\dfrac{1}{4} = 32\dfrac{3}{4}[/tex]

The distance between the drone and the fossil = [tex]32\dfrac{3}{4}[/tex] ft.

Therefore;

The distance between the drone and the fossil > 32 ft.

Answer:

a part

Step-by-step explanation:

Attachment added....

Hope it helps :)

Using the DINK method (Dual Income, No Kids), the total insurance need for Marianne and Roger is $172,000.

The DINK method calculates the insurance need based on the financial obligations and potential future expenses of a couple with dual incomes and no children. To determine their total insurance need, we consider their outstanding debts and potential expenses in case of death.

The calculations are as follows:

Mortgage: $125,000 (outstanding debt)

Car loans: $25,000 (outstanding debt)

Student loans: $22,000 (outstanding debt)

Funeral expenses: $12,000

The total outstanding debt and funeral expenses amount to $184,000.

However, the DINK method suggests subtracting the couple's annual incomes from the total insurance need since they have secure careers and can continue to generate income even in the event of the death of one spouse.

Both Marianne and Roger earn $45,000 annually, so we subtract $45,000 from the total, resulting in a total insurance need of $139,000.

Therefore, the correct answer is $172,000.

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Answer:

x=6√3

Step-by-step explanation:

In 30-60-90 triangles, if the hypotenuse is x, the longest leg is [tex]\frac{x}{2}[/tex], and the shortest leg is [tex]\frac{x\sqrt{3} }{2}[/tex].

If the longest leg is 6, then [tex]\frac{x}{2} =6[/tex]

Multiply by 2,

[tex]x=12[/tex]

If x=12,

[tex]\frac{12\sqrt{3} }{2}[/tex]

[tex]=6\sqrt{3}[/tex]

So, the shortest leg is 6√3.

The producer surplus at a price of $250 based on the supply curve, s(x) = 180 + 0.3·[tex]x^{\frac{3}{2} }[/tex] is about $1,326.5

The supply curve is the graphical representation of the price of a good or service and the quantity that manufacturers are willing to supply or provide.

The function for the supply curve is; s(x) = 180 + 0.3·[tex]x^{\frac{3}{2} }[/tex]

When the selling price is $250, the quantity of the product supplied can be found from the equation; s(x) = 250 = 180 + 0.3·[tex]x^{\frac{3}{2} }[/tex]

0.3·[tex]x^{\frac{3}{2} }[/tex] = 250 - 180 = 70

[tex]x^{\frac{3}{2} }[/tex] = 70/0.3 = 700/3

x = (700/3)[tex]\frac{2}{3}[/tex] ≈ 37.9

The quantity of soap supplied when the price is $250, is therefore, about 37.9

The producer surplus can be calculated from the area of a trapezoid formula as follows;

A = (b₁ + b₂) × h/2

Where the length of the bases are; b₁ ≈ 37.9,   b₂ = 0, and the height, h = (250 - 180) = 70

A = (37.9 + 0) × 70/2 = 1326.5

Therefore, at a selling price of $250, the producer surplus is about $1,326.5

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