Main Answer: The solution of the differential equation of y''(t) + 2y'(t) + y(t) = f(t) for t > 0 in terms of the error function is y(t) = f(t) * erf((t-sqrt(2t))/sqrt(2)) * e^-t - sqrt(2/π) * ∫0t [f(τ) * e^(-(t-τ)) * e^((τ-sqrt(2τ))/sqrt(2))] dτ.
Supporting Explanation:
The differential equation of y''(t) + 2y'(t) + y(t) = f(t) for t > 0 is a second-order linear ordinary differential equation with constant coefficients, where f(t) is the forcing function. To solve the equation, the homogeneous solution can be found by assuming that y(t) = e^rt. Substituting this into the differential equation and solving for the roots of the characteristic equation, gives the general solution of the homogeneous equation as y_h(t) = c_1e^(-t) + c_2te^(-t), where c1 and c2 are arbitrary constants.
To find the particular solution of the non-homogeneous equation, the method of undetermined coefficients can be used. However, if the forcing function is in the form of a Gaussian function, then it is more convenient to use the error function. The error function is defined as erf(x) = (2/√π) ∫0x e^(-t^2) dt, which has the properties of erf(-x) = -erf(x) and erf(x) = 1 - erf(-x).
The particular solution of the non-homogeneous equation can then be written as y_p(t) = f(t) * erf((t-sqrt(2t))/sqrt(2)) * e^-t. The complementary solution and the particular solution are added together to obtain the general solution of the non-homogeneous equation. However, due to the exponential function in the particular solution, the superposition principle does not apply and the integral of the product of f(τ) and the exponential function needs to be evaluated. This gives the complete solution of y(t) = y_h(t) + y_p(t) = c_1e^(-t) + c_2te^(-t) + y_p(t) = f(t) * erf((t-sqrt(2t))/sqrt(2)) * e^-t - sqrt(2/π) * ∫0t [f(τ) * e^(-(t-τ)) * e^((τ-sqrt(2τ))/sqrt(2))] dτ.
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Micah has been learning about Scientific Notation in math class and is frustrated because he doesn’t understand why he needs to learn it. While working on his homework with his mother, she told Micah that Scientific Notation can be very useful in certain careers. What do you think she meant?
Answer:
Step-by-step explanation:
Which graph is a function of x?
Answer:
The first graph is the function of x
Let f (x) = x2 + 2x + 3 What is the average rate of change for the quadratic function from X = -2 to x= to x = 5?
Answer:
Answer: 5
Step-by-step explanation:
The average rate of change for the function f(x) from x=a to x=b is given by :-
The given function : (Its a quadratic function with degree 2.)
The average rate of change for the quadratic function from x=−2 to x = 5 will be :-
[Note (-)(-)=(+)]
Hence, the average rate of change for the quadratic function from x=−2 to x = 5 is 5 .
Step-by-step explanation:
Solve the system of equations graphically: x – y = 1 and 3x = 3y + 3
Answer:
x=y+1,y=y
Step-by-step explanation:
Whats the value of x????
Answer:
X = 23
Step-by-step explanation:
oppsosite angles are equal so the brackets equal 90 degrees like the right angle opposite.
90-21 = 69
69 / 3 = 23
x = 23
Answer:
23
Step-by-step explanation:
The equation on the left will equal to 90 due to it being congruent to the angle with the square. Therefore, make the equation out to be 3x+21=90 and subtract 21 from both sides and that will give you 69. Divide 69 by three which will leave you with x=23
Triangle DEF ~ Triangle BCA
Select ALL angles that have a sine of 12/37
- A
- B
- C
- D
- E
- F
The two triangles, Triangle DEF ~ Triangle BCA. answer that DE = 3.6, EF = 1.47 and FD = 2.8.
Given the two triangles, Triangle DEF ~ Triangle BCA. Let us understand this notation first. '~' means 'is similar to'. So, it says that Triangle DEF is similar to Triangle BCA. The meaning of similarity is that the two triangles have the same shape but different sizes,
Similarly, angles E and C, as well as angles F and A, are corresponding angles. Since corresponding angles of similar triangles are congruent, we can write:D = B, E = C and F = ABy the transitive property of equality, we can write:D = B, E = C, F = A => DE = BC, EF = AB and FD = CA (using corresponding sides of the triangles).So, the triangles DEF and BCA are similar and their corresponding sides are proportional. Therefore, we can write: DE/BC = EF/AB = FD/CA = k (some constant)If we consider DE/BC = k, we can write: DE/k = BC and DE/k + EF/k = AB and (DE + EF)/k = FDAnd, from the given information, we know that CD = 3, EA = 4 and FB = 6.
So, we can write:BC + CD = 5 (since AB = EA + FB = 4 + 6 = 10 and AB/BC = EA/CD => BC = AB * CD/EA = 10 * 3/4 = 7.5)FD + CD = 9 (since FD/CA = EF/EA => FD = EF * CA/EA = 6 * 9/4 = 13.5)So, we get the following system of equations:DE/k + EF/k = 10/k = 2.5 (from the given EF = 2.5)DE/k = 7.5 - CD = 4.5 (from the equation BC + CD = 5)EF/k = AB - DE/k = 5.5 (using the first equation)DE/k + FD/k = 13.5/k (from the equation FD + CD = 9)DE/k = CD = 3 (from the given information)FD/k = 10.5/k (using the third equation)Solving these equations, we get:k = 3.75, DE = 13.5/3.75 = 3.6, EF = 5.5/3.75 = 1.47 and FD = 10.5/3.75 = 2.8.
So,we get the answer that DE = 3.6, EF = 1.47 and FD = 2.8.
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The master equation describing the evolution of the probability Pm (t) to find an electron on site m of a linear chain of molecules with lattice constant a = 1nm is given by: dt -2RP.m + RPm+1 + RPm-1 Assuming that R = 10125-1, find the diffusion constant in cm²/s. Estimate a characteristic time for the electron to move a distance of 1 micron.
The electron to move a distance of 1 micron is approximately 0.255 s.
The main equation describing the evolution of the probability Pm (t) of finding an electron at point m in a molecular chain with lattice constant a = 1 nm is given by the formula: dt = -2RP.m RPm 1 RPm- 1 Assuming that R = 10-125 , calculate the diffusion constant in units of cm²/s. The diffusion constant is given by D = a²v/2t where a = 1 nm, v = RPm = 2RP and R = 10-125D = a²v/2tD = (10-9)² x 2RP / 2t = 4.2 x 10 ⁻¹⁰ RPt
Evaluate the characteristic time for an electron to travel a distance of 1 micron. Room temperature can be assumed for the electron, T = 27°C = 300K. The diffusion constant is given by the formula D = kbT/6πηra = (1.38 x 10-23 J/K) (300K) / (6π(8.9 x 10-4) Nsm-² x 1 x 10-⁷ m)D = 1 .57 x 10-2 cm²/s Distance 1 micron = 10⁻⁴ cm So by the formula D = sqrt((2d²)/t) t = 2d²/Dt = 2 x (10⁻⁴)² / (1.57 x 10 ⁻⁵ ) = 0.255 s Therefore, the characteristic time of an electron to travel a distance of 1 micron is approximately 0.255 s.
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Mark hopes to one day earn $10,000. Estimate how
many clients Mark would need.
I need help in this. Please
Answer:
search it up :)
Step-by-step explanation:
Find the interest on the loan, $35,000 at 6% for 9 months. A. $1,910 C. $1,575 B. $1,395 D. $1,465
The interest on the loan of $35,000 at a 6% interest rate for 9 months is $1,575.
To calculate the interest on a loan, you can use the formula:
Simple Interest = Principal x Rate x Time.
In this case, the principal is $35,000, the interest rate is 6%, and the time is 9 months.
First, convert the interest rate to a decimal by dividing it by 100: 6% = 0.06. Next, calculate the interest by multiplying the principal, rate, and time: Interest = $35,000 x 0.06 x 9/12.
Simplifying the calculation, we get: Interest = $35,000 x 0.06 x 0.75 = $1,575.
Therefore, the correct answer is option B: $1,575.
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Select all the expressions that are equivalent to 4 – x. x – 4 4 + -x -x + 4 -4 + x 4 + x Use the distributive property to write an expression that is equivalent to 5(-2x – 3). If you get stuck, use the boxes to help organize your work.
Answer:
B
Step-by-step explanation:
Answer:
B
Step-by-step explanation:
What is the answer. Please do NOT send me a link. I am severely struggling with this question.
Answer:
360 ft²
³
Step-by-step explanation:
The volume of the rectangular pyramid is calculated as
V = [tex]\frac{1}{3}[/tex] Ah ( A is the area of the base and h the perpendicular height )
Here A = 10 × 6 = 60 ft² and h = 6 , then
V = [tex]\frac{1}{3}[/tex] × 60 × 6 = 20 × 6 = 120 ft³
The volume of the rectangular prism is calculated as
V = 10 × 6 × 4 = 60 × 4 = 240 ft³
Total volume = 120 + 240 = 360 ft³
Lind domain and Range for 9x) = 1-dx²_X+2 without graphing. Prove that [0, 1] & [11, 21]
The domain of the function is all real numbers, and the range is [0, 1] and [11, 21].
To find the domain and range of the function f(x) = 1 - dx² - x + 2, we can analyze its properties without graphing.
Domain:
The domain of a function consists of all the valid input values, or the values of x for which the function is defined. In this case, the only restriction we need to consider is the square root of a negative number.
In the given function, there are no square roots involved. Therefore, the function is defined for all real numbers. The domain is the set of all real numbers, which can be represented as (-∞, ∞).
Range:
The range of a function consists of all the possible output values, or the values that f(x) can take. To determine the range, we need to consider the behavior of the function as x approaches positive or negative infinity.
As x approaches positive or negative infinity, the dominant term in the function is -dx². If d is positive, as x gets larger, the term -dx² becomes more negative, approaching negative infinity. Similarly, if d is negative, as x gets larger, the term -dx² becomes more positive, approaching positive infinity.
Since the remaining terms in the function (1-x+2) are constants, they do not affect the behavior as x approaches infinity. Therefore, the range of the function depends on the value of d.
If d is positive, the range of the function is (-∞, 1+d). As x approaches negative infinity, the function approaches positive infinity, and as x approaches positive infinity, the function approaches 1+d.
If d is negative, the range of the function is (1+d, ∞). As x approaches negative infinity, the function approaches 1+d, and as x approaches positive infinity, the function approaches negative infinity.
Given that the range is [0, 1] & [11, 21], we can conclude that d is positive, and the range is [0, 1+d]. Since the range also includes [11, 21], we can infer that 1+d = 11, and solving for d gives d = 10.
Therefore, the domain of the function is (-∞, ∞), and the range is [0, 1] & [11, 21].
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When working in trigonometry, you typically measure angles in
Answer:
Radians
Hope that helps!
Step-by-step explanation:
Can someone help me asap please.
How do you do this problem?
Answer:
Step-by-step explanation:
0.3a = -6
Divide 0.3 from both sides
a = -20
HELP ASAP 20 POINTS BRAINLIEST
Answer:
56.55 [tex]cm^{3}[/tex]
Step-by-step explanation:
r = 6/2 = 3
Area of a cylinder:
π[tex]r^{2}[/tex] x length
[tex]3^{2}[/tex]π x 6 = 54π
Area of sphere:
[tex]\frac{4}{3}[/tex] x π x [tex]r^{3}[/tex]
[tex]\frac{4}{3}[/tex] x π x [tex]3^{3}[/tex] = 36π
54π - 36π = 18π = 56.55
4,14,14, 14.4.4
Determine if the data set is unimodal, bimodal, multimodal, or has no mode. Identify the models). If any exist
The data set is bimodal with modes at 4 and 14.
To determine the mode(s) and the modality of a data set, we need to identify the values that occur most frequently.
The given data set is: 4, 14, 14, 14, 4
To find the mode(s), we can count the frequency of each value:
4 appears 2 times14 appears 3 times
The mode(s) are the value(s) that appear with the highest frequency. In this case, both 4 and 14 have the same frequency of occurrence, so this data set is bimodal, with modes at 4 and 14.
Therefore, the data set is bimodal with modes at 4 and 14.
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evaluate plssssssssssssssssssss
Answer:
2/5!
Step-by-step explanation:
The cone below has a radius of 1 inch and height of 4 inches. What is the slant height in inches?
a. √-5
b. √−−17
c. 15
d. 17
Answer:
B is the correct answer ([tex]\sqrt{-17}[/tex])
Step-by-step explanation:
The slant height of the cone is √17.
What is Cone?A cone is a shape formed by using a set of line segments or the lines which connects a common point, to all the points of a circular base.
Here, radius = 1 inch
height = 4 inches
Slant height = [tex]\sqrt{r^2+h^2}[/tex]
= [tex]\sqrt{1^2+4^2}[/tex]
= [tex]\sqrt{1+16}[/tex]
l = √17 inches.
Thus, the slant height of the cone is √17.
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Find the volume of the cone.
please help me.
Answer:
46.08
Step-by-step explanation:
the formula for finding the volume of the cone is: V = π r^2 h/3
Insert the information provided (2 because half the diameter is the radius)
and solve. Hope this helps!
please answer 100 pt
Answer:
corresponding angle are:
<1=<5
<2=<6
<7=<3
<8=<4
<1&<5,<2&<6,<7&<3,<8&<4are pair corresponding angle.
Answer:
Step-by-step explanation:
corresponding angles are two angles at the same point on two parallel lines cut by a transversal.
so 1 n 5 are corresponding angles
so are 2 n 6, 3 n 7 & 4 n 8
You poll your friends on how many different states they have visited and plot the results in a histogram. Which of the following cannot be the value of SS (sum of squares) for the data you collected, and why? | Select] Thinking of those same data (how many states your friends have visited), which of the following cannot be the value of the variance and standard deviation, and why? (
The sum of squares cannot be D. -25, because SS can never be negative.
We know that the sum od squares is
∑(xₐ - mean)²
where,
xₐ are the data point recorded. The sum of squares measures the deviation of the data points from the mean of the variable. The sum of squares can be total, egressive, or residual.
Hence in a regression analysis of an equation
Y = A + BX + E
where X is the explanatory variable, E is the error term or residual and Y is the explained variable, we can get a sum of squares of Y, X, and E.
the sum of squares can be 0 if there is no variation of the data points from the mean, or even a very large number if the variation is high.
Since the numbers are essentially squared, the only way to get a negative sum of squares is if imaginary numbers are involved, which is an impossible task in regression analysis as it deals with real data. Hence the sum of squares can never be negative.
Hence from available options, the sum of squares cannot be D. -25, because SS can never be negative
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Complete Question
You poll your friends on how many different states they have visited and plot the results in a histogram. Which of the following cannot be the value of SS ( sum of squares) for the data you collected, and why?
A. 1, because its too low
B. 144, because there arent that many states
C. 3.87, bwcause you cant hahe a decimal
D. -25, because SS can never be negative
in february 2002 the argentine peso lost 70% of its value compared to the united states dollar. this devaluation drastically raised the price of imported products. according to a survey conducted by ac nielsen in april 2002, 68% of the consumers in argentina were buying fewer products than before the devaluation, 24% were buying the same number of products, and 8% were buying more products. furthermore, in a trend toward purchasing less-expensive brands, 88% indicated that they had changed the brands they purchased. f an appetizer, entree, beverage, and dessert. you have a choice of five appetizers, ten entrees, three beverages, and six desserts. how many possible complete dinners are possible?
After considering the given data we conclude that the there are 900 possible complete dinners generated that will be satisfactory to the dedicated question.
To evaluate the number of possible complete dinners, we need to apply multiplication regarding the number of choices for each course.
The number of possible complete dinners is the product of the number of choices for the appetizer, entree, beverage, and dessert.
Its is known to us that we have a choice of five appetizers, ten entrees, three beverages, and six desserts, the number of possible complete dinners is:
Number of possible complete dinners = number of choices for appetizer × number of choices for entree × number of choices for beverage × number of choices for dessert
Number of possible complete dinners = 5 × 10 × 3 × 6
Number of possible complete dinners = 900
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The complete question is
A restaurant menu has a price-fixed complete dinner that consists of an appetizer, an entrée, a beverage, and a dessert. You have a choice of 5 appetizers, 10 entrées, 3 beverages, and 6 desserts. Determine the total number of possible dinners.
When approximating S(x)dx using Romberg integration, R9,4 gives an approximation of order: h10 h8 h4 h6
R₄,₄ gives an approximation of order h⁸ when approximating ∫(a to b)f(x)dx using Romberg integration. Therefore second option is the correct answer.
When approximating the integral ∫(a to b) f(x) dx using Romberg integration, the term "R₄,₄" refers to the fourth row and fourth column of the Romberg matrix.
This specific entry represents the approximation obtained using the Romberg method with four iterations. The order of approximation is determined by the highest power of h in the error term of the approximation.
Since R₄,₄ has a subscript of 4, it indicates that the approximation is of order h⁸. This means that the error decreases at a rate of h⁸ as the step size h decreases, providing a more accurate estimation of the integral.
Therefore the correct answer is second option.
The question should be:
When approximating ∫(a to b)f(x)dx using Romberg integration, R₄,₄ gives an approximation of order:
h10
h8
h4
h6
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Round off 3,489 to the nearest hundred.
Answer:
3,500
plz mark me as brainliest
At the movies josh buys somes snacks for his family. He purchases one bucket of popcorn for$ 9.50 and four drinks. If he spent $20.50 total, write and solve an equation to find d, the amount each drink cost.
Answer:
$2.75
Step-by-step explanation:
9.50-20.50=11.00
11.00÷4=2.75
Answer:
2.75; 9.50 + 4x = 20.50
Step-by-step explanation:
First, lets write an equations: 9.50 + 4x = 20.50. Subtract 9.50 from 20.50, giving us: 4x = 11. Then divide 11 by 4, giving us x = 2.75. Each drink cost $2.75. To check this, plug in 2.75 for x in the original equations and you will get 20.50.
Luis paid a total of $72.00 for 3 pairs of shorts. If each pair of shorts costs the same amount, what is the price of one pair of shorts?
Answer:
24
Step-by-step explanation:
72/3
Problem 5 Let X₁.....Xy be iid according to EX= for all i=1,2,.... Define Y = 1 if X, > density function of Y₁. a continuous probability density fx. Suppose and Y₁ = 0 otherwise. Find the probability
Given that X1, X2, ..., Xy are i.i.d according to the probability density function fx such that EX = 2. Define Y = 1 if X > 2 and Y = 0 otherwise.
Find P(Y = 1).
Given that X1, X2, ..., Xy are i.i.d according to the probability density function fx such that EX = 2. Thus the probability density function is,fx = 1/2 for x ∈ (0, 2) fx = 0 elsewhere
Therefore, P(X > 2) = ∫2∞ fx dx= ∫2∞ 0 dx= 0Now, P(Y = 1) = P(X > 2) = 0
Hence, the required probability is 0. Therefore, the correct option is (D) 0.
Likelihood is a proportion of the probability of an occasion to happen. Numerous occurrences cannot be completely predicted. Using it, we can only predict the chance of an event happening, or how likely it is to happen.
The probability of an event occurring is determined by dividing the total number of possible outcomes by the number of favorable outcomes. A coin flip is the simplest illustration. There are only two outcomes that can occur when flipping a coin; either heads or tails is the outcome.
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I REALLY NEED HELP IF YOU CAN PLEASE SHOW WORK SO I CAN UNDERSTAND
Answer:
G
Step-by-step explanation:
The number of total bottles sold on that day is 40
11 + 7 + 18 + 4 = 40
On Tuesday, theoretically, this is what will be sold:
22 bottles of apple juice
14 bottles of cranberry juice
36 bottles of orange juice
8 bottles of pineapple juice
We need to find out what doesn't fit the data.
F: 14 - 8 = 6, not correct answer
G: the number of cranberry juice will be 6 times the number of pineapple juice sold? It's not even twice the amount!
H: 36 + 14 = 50, not correct answer
J: 22 - 8 = 14, not correct.
From this, G is the correct answer as it is not even close to matching the data.