Problem 2. (6 points total) Consider the following IVP for some constant k> 0. dy dt + ky = cos(vk+1.t) ( y(0) = 0 (y'(0) = 0 (a) (3 points) Show the work required to solve this IVP by hand. Your solu
Answers
To solve the given initial value problem (IVP) by hand, we'll follow these steps: Step 1: Write the differential equation. The given differential equation is: dy/dt + ky = cos((vk+1)t).
Step 2: Identify the integrating factor. The integrating factor is given by the exponential of the integral of the coefficient of y, which is k in this case: IF = e^(∫ k dt) = e^(kt). Step 3: Multiply the differential equation by the integrating factor. Multiplying both sides of the equation by the integrating factor, we get: e^(kt) * (dy/dt) + e^(kt) * ky = e^(kt) * cos((vk+1)t). Step 4: Apply the product rule to simplify the left side. Using the product rule for differentiation on the left side, we have:(d/dt)(e^(kt) * y) = e^(kt) * cos((vk+1)t). Step 5: Integrate both sides: Integrating both sides of the equation with respect to t, we get: ∫ (d/dt)(e^(kt) * y) dt = ∫ e^(kt) * cos((vk+1)t) dt. The left side simplifies to: e^(kt) * y
For the right side, we can integrate by parts to handle the product of functions: ∫ e^(kt) * cos((vk+1)t) dt = (1/k) * e^(kt) * sin((vk+1)t) - (v+1)/k * ∫ e^(kt) * sin((vk+1)t) dt. Step 6: Simplify the integral on the right side. To evaluate the integral ∫ e^(kt) * sin((vk+1)t) dt, we can use integration by parts again. Let's define u = e^(kt) and dv = sin((vk+1)t) dt. Then, we have du = k * e^(kt) dt and v = -(v+1)/((vk+1)^2 + 1) * cos((vk+1)t). Using the formula for integration by parts: ∫ u dv = uv - ∫ v du. Applying this formula, we get: ∫ e^(kt) * sin((vk+1)t) dt = - (v+1)/((vk+1)^2 + 1) * e^(kt) * cos((vk+1)t) - k/((vk+1)^2 + 1) * ∫ e^(kt) * cos((vk+1)t) dt. Step 7: Substitute the integral back into the equation. Substituting the integral back into the original equation, we have: e^(kt) * y = (1/k) * e^(kt) * sin((vk+1)t) - (v+1)/k * ((v+1)/((vk+1)^2 + 1) * e^(kt) * cos((vk+1)t) + k/((vk+1)^2 + 1) * ∫ e^(kt) * cos((vk+1)t) dt)
Step 8: Solve for y. Now, we can cancel out the common factors of e^(kt) on both sides and solve for y. Finally, we apply the initial conditions y(0) = 0 and y'(0) = 0 to determine the specific values of the constant v and solve for the constant k. Note: Due to the complexity of the calculations involved, it would be more efficient to use numerical methods or software to solve this IVP and determine the values of v and k.
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Related Questions
i^25
explain please
Answers
Answer:
Step-by-step explanation:
1.step. i^25^i
2.step. Rewrite (i^4)6^i=i^4
3.step. Rewrite (i^2)^2
4.step. ((i^2)^2)^6i
5.step. i^2 as -1
6.step. ((-1)^2)^6i
7.step. Raise −1 to the power of 2
i^6i
8.step.One to any power is one.
i1
9.step.Multiply i by 1.
ix1=i
Find the compound interest on Rs 8000 for 1 year at 20% per annum , when the interest is compounded half yearly.
Answers
PLEASE HELP! WILL CHOSE BRAINLIST!
Suppose you are making a scale drawing Find the length of each object on the scale drawing with given scale. Then find the scale factor. 5-6 a parking lot 480 meters wide; 1 centimeter= 16.5 meters 7-8 A desk 6 feet long; 1.5 inches= 0.5 feet. Sorry the first ones that I sent were not finished.
Answers
Answer:
1 cm = 16.5 m
1 m = 1 : 16.5 = 0.060606 cm
480 * 0.060606 = 29.09088 cm ( on the scale drawing )
B :
1.5 in = 0.5 ft
6 ft = 0.5 ft * 12
12 * 1.5 in = 18 inches
Step-by-step explanation:
Describe the translation of the point to its image. (-2,-3) to (-2,4)
Answers
Up 8 Units, Hope this helps :)
write the equation of the line in fully simplified slope-intercept form?
Answers
Answer:
y= -2x+1
Step-by-step explanation:
i hope this helps :)
Good luck hope it helps :)
Which of the following is true of a rectangle, but may not be true of a parallelogram.
The opposite angles are congruent.
The opposite sides are parallel.
The diagonals bisect each other.
The diagonals are congruent.
Answers
Answer:
Diagonals are congruent
Step-by-step explanation:
In rectangle, diagonals are congruent.
But in parallelogram diagonals are not congruent
What is the simplified value of the exponential expression 273?
不
1
Ο Ο Ο
9
Answers
Answer:
The answer is 3
For the steps, Im leaving it blank for you to figure it out first.
If you really need help, ask anytime and I'll reply as soon as i can :)
PLEASE HELP MEEE!! NEED HELP FAST!! 20 POINTS
the current balance in Marianne's saving account is $525. The account pays 3.5% compounded annually. If she makes no deposits or withdrawals for the next 4 years,her savings account balance will be-
Please explain with the correct interest formula.
Answers
30000$ swhat i thiiiiiiiiiiiiiiiiiiiin
===========================================
Explanation:
The formula we'll use is
A = P(1+r/n)^(n*t)
which is the compound interest formula
The variables are
P = amount deposited (principal) = 525r = interest rate in decimal form = 0.035n = compounding frequency = 1 (annually means 1 time a year)t = number of years = 4So we get
A = P(1+r/n)^(n*t)
A = 525(1+0.035/1)^(1*4)
A = 602.449575328124
A = 602.45
Which is her balance after four years assuming no additional deposits or withdrawals.
------------
Extra info:
Subtract off the initial balance to get 602.45 - 525 = 77.45
She made 77.45 dollars in interest over those four years.
FILL IN THE BLANK. "Test Yourself
If a and b are integers, the notation a | b denotes ________________ and the notation a / b denotes __________________ ."
Answers
If a and b are integers, the notation a | b denotes that a divides b, or in other words, that b is a multiple of a. The symbol "|" is read as "divides." For example, we write 2 | 6, which means that 2 divides 6 since 6 is a multiple of 2.
On the other hand, the notation a / b denotes the quotient of a divided by b, or in other words, the result of dividing a by b. The symbol "/" is read as "divided by." For example, we write 6 / 2 = 3, which means that the quotient of 6 divided by 2 is equal to 3.
It is important to note that the notation a | b and a / b are not equivalent. The former is a statement about the divisibility of two integers, while the latter is a numerical expression representing the result of dividing one integer by another.
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What is the solution to the system of linear equations shown? picture attached
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For the quadrilateral on the grid, the slope of WX is 0, the slope of XY is , the slope of YZ is 0, and the slope of ZW is .
Which statement verifies that quadrilateral WXYZ is a parallelogram?
The difference of the slope of adjacent sides is .
The slopes of adjacent sides are opposite reciprocals.
The opposite sides have the same slope.
XY and YZ have the same slope.
Answers
The statement that verifies that quadrilateral WXYZ is a parallelogram is C)The opposite sides have the same slope.
What is a Parallelogram?This refers to a four-sided figure that is used in geometry and has parallel opposite sides.
Hence:
Remember that the opposite sides of a parallelogram are parallel.
We would solve:
Slope WX = 0
Slope XY= 5/2
Slope XZ= 0
Slope ZW= 5/2
The next step would be:
Slope WX= Slope YZ
Slope XY = Slope ZW
Based on the fact that the opposite sides have the same slopes, we can conclude that quadrilateral WXYZ is a parallelogram
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What is the slope of the line through (-10,1) and (0,-4) ?
Answers
Answer:
-5/10 or -1/2
Step-by-step explanation:
Write tan 41π/36 in terms of the tangent of a positive acute angle.
Answers
tan(41π/36) can be written in terms of the tangent of a positive acute angle as (tan((1/9)π) + tan((37/36)π)) / (1 - tan((1/9)π)tan((37/36)π))
To express tan(41π/36) in terms of the tangent of a positive acute angle, we need to find an angle within the range of 0 to π/2 that has the same tangent value.
First, let's simplify 41π/36 to its equivalent angle within one full revolution (2π):
41π/36 = 40π/36 + π/36 = (10/9)π + (1/36)π
Now, we can rewrite the angle as:
tan(41π/36) = tan((10/9)π + (1/36)π)
Next, we'll use the tangent addition formula, which states that:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
In this case, A = (10/9)π and B = (1/36)π.
tan(41π/36) = tan((10/9)π + (1/36)π) = (tan((10/9)π) + tan((1/36)π)) / (1 - tan((10/9)π)tan((1/36)π))
Now, we need to find the tangent values of (10/9)π and (1/36)π. Since tangent has a periodicity of π, we can subtract or add multiples of π to get equivalent angles within the range of 0 to π/2.
For (10/9)π, we can subtract π to get an equivalent angle within the range:
(10/9)π - π = (1/9)π
Similarly, for (1/36)π, we can add π to get an equivalent angle:
(1/36)π + π = (37/36)π
Now, we can rewrite the expression as:
tan(41π/36) = (tan((1/9)π) + tan((37/36)π)) / (1 - tan((1/9)π)tan((37/36)π))
Since we are looking for an angle within the range of 0 to π/2, we can further simplify the expression as:
tan(41π/36) = (tan((1/9)π) + tan((37/36)π)) / (1 - tan((1/9)π)tan((37/36)π))
Therefore, tan(41π/36) can be written in terms of the tangent of a positive acute angle as the expression given above.
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Enter the missing value needed to find the length of CB make sure to simplify your answer
Answers
6b
We are given the distance formula on the bottom in red which we can use to find the distance of CB. The formula tells us that we first need to substract the x-values of the two points and the y-values of the two points. When we substract the x-values of the two points, we get 4a, as given in the problem, the calculation would be as follows:
3a - (-a) = 3a + a = 4a
Now, we need to find the difference between the y-values in the same way, by substracting y1 from y2. In this case, y2 refers to the y-coordinate of point B while y1 refers to the y-coordinate of point C. The calculation is as follows:
b - (-5b) = b + 5b = 6b
Our final answer is 6b
Can someone please show me how to solve this?
Jane walks 5.0 miles in the southwest direction and then 8.0 miles in the direction 70 degree north of west. What is the final displacement of Jane in magnitude and direction?
Answers
The final displacement of Jane is approximately 11.281 miles in the direction of approximately 88.8 degrees clockwise from the positive x-axis.
To solve this problem, we can use vector addition to find the final displacement of Jane.
Step 1: Determine the components of each displacement.
The southwest direction can be represented as (-5.0 miles, -45°) since it is in the opposite direction of the positive x-axis (west) and the positive y-axis (north) by 45 degrees.
The direction 70 degrees north of the west can be represented as (8.0 miles, -70°) since it is 70 degrees north of the west direction.
Step 2: Convert the displacement vectors to their Cartesian coordinate form.
Using trigonometry, we can find the x-component and y-component of each displacement vector:
For the southwest direction:
x-component = -5.0 miles * cos(-45°) = -3.536 miles
y-component = -5.0 miles * sin(-45°) = -3.536 miles
For the direction 70 degrees north of west:
x-component = 8.0 miles * cos(-70°) = 3.34 miles
y-component = 8.0 miles * sin(-70°) = -7.72 miles
Step 3: Add the components of the displacement vectors.
To find the total displacement, we add the x-components and the y-components:
x-component of total displacement = (-3.536 miles) + (3.34 miles) = -0.196 miles
y-component of total displacement = (-3.536 miles) + (-7.72 miles) = -11.256 miles
Step 4: Find the magnitude and direction of the total displacement.
Using the Pythagorean theorem, we can find the magnitude of the total displacement:
\(magnitude = \sqrt{(-0.196 miles)^2 + (-11.256 miles)^2} = 11.281 miles\)
To find the direction, we use trigonometry:
direction = atan2(y-component, x-component)
direction = atan2(-11.256 miles, -0.196 miles) ≈ -88.8°
The final displacement of Jane is approximately 11.281 miles in the direction of approximately 88.8 degrees clockwise from the positive x-axis.
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find teh exact value of sin 2x given that sec x = 3/2 and csc y = 3 and x and y are in quadrant 1
Answers
The exact value of \(sin 2x\) is \(4√5/9.\)
Given that \(sec x = 3/2 and csc y = 3\)where x and y are in the 2x = 2 sin x quadrant, we need to find the exact value of sin 2x.
In the first quadrant, we have the following values of the trigonometric ratios:\(cos x = 2/3 and sin y = 3/5\)
Also, we know that sin \(2x = 2 sin x cos x.\)
Now, we need to find sin x.
Having sec x = 3/2, we can use the Pythagorean identity
\(^2x + 1 = sec^2xtan^2x + 1 = (3/2)^2tan^2x + 1 = 9/4tan^2x = 9/4 - 1 = 5/4tan x = ± √(5/4) = ± √5/2\)
As x is in the first quadrant, it lies between 0° and 90°.
Therefore, x cannot be negative.
Hence ,\(tan x = √5/2sin x = tan x cos x = √5/2 * 2/3 = √5/3\)
Now, we can find sin 2x by using the value of sin x and cos x derived above sin \(2x = 2 sin x cos xsin 2x = 2 (√5/3) (2/3)sin 2x = 4√5/9\)
Therefore, the exact value of sin 2x is 4√5/9.
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Use the point-slope form to find the
equation given the point and slope
m= -1/2 and (-4,6)
Answers
Answer:
y = -½x + 4
Step-by-step explanation:
Given a slope or m of -1/2, and a point (-4,6).
point slope is y-y1 = m(x-x1).
under what circumstances is the point-biserial correlation used?
Answers
Answer: when one of the variables is dichotomous; when you need to measure the relationship between a continuous variable and a dichotomous variable.
Step-by-step explanation:
Do the following using the given information: Utility function u(x1+x2) = .5ln(x1) + .25ln(x₂) .251 Marshallian demand X1 = - and x₂ = P₂ . Find the indirect utility function . Find the minimum expenditure function . Find the Hicksian demand function wwww
Answers
Hicksian demand functions are:x1** = 2P₁x₂ ; x₂** = P₂²
Utility function: u(x1+x2) = .5ln(x1) + .25ln(x₂) .The Marshallian demand functions are: x1* = - and x₂* = P₂.
The indirect utility function is found by substituting Marshallian demand functions into the utility function and solving for v(P₁, P₂, Y).u(x1*,x2*) = v(P₁,P₂,Y) ⇒ u(-, P₂) = v(P₁,P₂,Y) ⇒ .5ln(-) + .25ln(P₂) = v(P₁,P₂,Y) ⇒ v(P₁,P₂,Y) = - ∞ (as ln(-) is not defined)
Thus the indirect utility function is undefined.
Minimum expenditure function can be derived from the Marshallian demand function and prices of goods:
Exp = P₁x1* + P₂x2* = P₁(-) + P₂P₂ = -P₁ + P₂²
Minimum expenditure function is thus:
Exp = P₁(-) + P₂²
Hicksian demand functions can be derived from the utility function and prices of goods:
H1(x1, P1, P2, U) = x1*H2(x2, P1, P2, U) = x2*
Hicksian demand functions are:
x1** = 2P₁x₂
x₂** = P₂²
If there are no restrictions on the amount of money the consumer can spend, the Hicksian demand functions for x1 and x2 coincide with Marshallian demand functions.
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Zara and her neighbor both had a neighborhood lemonade and cookie sale last weekend. Zara made $21.75 while her neighbor made $43.50.
Write a system of equations for this situation. Determine how much each cup of lemonade cost.
Answers
The cost of a lemonade is $1.5 while that of a cookie is $0.75.
EquationAn equation is an expression used to show the relationship between two or more variables and numbers.
Let x represent the cost of a lemonade and y represent the cost of a cookie, hence:
9x + 11y = 21.75 (1)
Also
22x + 14y = 43.50 (2)
From equation 1 and 2:
x = 1.5, y = 0.75
The cost of a lemonade is $1.5 while that of a cookie is $0.75.
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10. | 23 | + | 19 | = I 19 I =
Answers
6th grade math, Please Help) 2 people pls answer so i can make brainliest! and you will get 20 points :)
Evaluate the Algebraic expression; j+2h= when f=6, g=8, h=12, j=2
OA) 14
OB)18
OC)26
OD)30
Answers
Answer:
the answer is 26
Step-by-step explanation:
j+2h
(2) + 2(12)
2 + 24
26
Use quadratic formula to solve 16+8x+x^2=0
Answers
Answer: x=-4
Step-by-step explanation:
As shown in the figure below, Luis walks 40 m toward his truck. He notices he forgot his keys, so he returns
back to his house. His total travel time is 95 s.
40 m
40 m
What is Luis's average velocity over the 95 s period?
Answers
Answer:
Velocity = 0
Speed = 0.84
Step-by-step explanation:
Khan Academy
Luis's average velocity over the 95 sec period is 0 m/s.
Given that, Luis walks 40 m toward his truck. He notices he forgot his keys, so he returns back to his house.
What is the average velocity?Magnitude of average velocity is always less than or equal to the average speed because displacement is always smaller than or equal to distance.
Average Velocity = Total time / Total Displacement
It is a vector quantity and has units of m/s.
Here, total time = 95 sec and displacement = 0 meter
Now, average velocity = 95/0
= 0 m/s
Therefore, Luis's average velocity over the 95 sec period is 0 m/s.
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A piling for a high-rise building is pushed by two bulldozers at exactly the same time. One bulldozer exerts a force of 1850
pounds in a westerly direction. The other bulldozer pushes the piling with a force of 3250 pounds in a northerly direction. What
is the direction of the resulting force upon the piling, to the nearest degree?
Answers
Answer: I got this answer from the internet hopefully this helps
Resultant force will act on 65° N of W.
Step-by-step explanation:
The direction of the resulting force upon the piling, to the nearest degree is gotten as; A = 60°
What is the direction of the resultant force?As given in the question one bulldozer exerts force of 1850 pounds in western direction and another pushes with a force of 3250 pounds in northern direction.
Now, we have to find angle A which is an angle between 1850 pounds of force and resultant force F.
Thus;
tan A= Force on North direction/ Force on west direction
tan A = 3250/1850
tan A = 1.7568
A = tan⁻¹1.7568
A = 60.35°
To the nearest degree is A = 60°
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The population of a South America country is about 17,400,000. Which expression shows this number in scientific notation?
Answers
Answer:
1.74x10^7
Step-by-step explanation:
you count how many times you moved the decimal to the right
can i have brainliest? i only need 2 more
Answer:
1.74x10^7
Step-by-step explanation:
2. Which of the following values of x make the following functions equal? y = 2x-1 y = -5x + 13
O x = 1
Ox=2
x = 3
Ox=4
Answers
Answer:
x=2
Step-by-step explanation:
y=2x-1
y=-5x+13
both y has the same value so,
2x-1=-5x+13
7x=14
x=2
Rewrite the linear function 7x−y=15 in slope-intercept form.
y=−7x−15
y=7x+15
y=−7x+15
y=7x−15
please help!!
Answers
What are two strategies the speaker uses to develop the point that voluntour opportunities often do more harm than good?expert testimonyfictional storiesrepetitionrhetorical questionstatistics
Answers
The speaker uses expert testimony and statistics to develop the point that voluntour opportunities often do more harm than good.
How does the speaker use the 2 strategies to develop the point?We have a speech that is intended to express personal opinions and to conceive the audience about the major issue in our case, which is that volunteer opportunities frequently cause more harm than good. Through expert testimony, the speaker uses a statement made by a subject-matter expert to bolster the main argument of the speech. Statistics demonstrating that there have been studies done to substantiate what has been asserted increase the credibility of arguments.The study of statistics is the field that deals with the gathering, structuring, analysing, interpreting, and presenting of data. In order to apply statistics to an issue in science, business, or society, it is customary to start with a statistical population or a statistical model that will be investigated.To learn more about statistics, refer:
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what is the solubility of pbf₂ in a solution that contains 0.0450 m pb²⁺ ions? (ksp of pbf₂ is 3.60 × 10⁻⁸)
Answers
Hi! The solubility of PbF₂ in a solution (Ksp =3.60 × 10⁻⁸) containing 0.0450 M Pb²⁺ ions is 2.83 × 10⁻⁴ M F⁻ ions.
To find the solubility of PbF₂ in a solution containing 0.0450 M Pb²⁺ ions, you can follow these steps:
1. Write the balanced equation for the dissolution of PbF₂:
PbF₂(s) ⇌ Pb²⁺(aq) + 2F⁻(aq)
2. Write the Ksp expression for PbF₂:
Ksp = [Pb²⁺][F⁻]²
3. Substitute the given Ksp value and the concentration of Pb²⁺ ions:
3.60 × 10⁻⁸ = (0.0450)[F⁻]²
4. Solve for the concentration of F⁻ ions:
[F⁻]² = (3.60 × 10⁻⁸) / 0.0450
[F⁻]² = 8.00 × 10⁻⁷
[F⁻] = 2.83 × 10⁻⁴ M
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