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Solve the differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y'(0) = 0.
y'' + 2y' - 8y = 4e^(-2x) - e^(-x)
y(x) =
Answers
The differential equation is y(x) = (1/10)e^(2x) + (3/5)e^(-x) - (1/10)e^(4x) - (2/5)x*e^(-x)
The differential equation: y'' + 2y' - 8y = 3e^(2x) - 2e^(-x)
Find the general solution to the homogeneous equation
The characteristic equation is r^2 + 2r - 8 = 0. Solving this equation gives the roots r1 = 2 and r2 = -4.
So the general solution to the homogeneous equation is y_h(x) = c1e^(4x) + c2e^(-2x).
Find the particular solution using variation of parameters
Let y_p(x) = u1(x)e^(4x) + u2(x)e^(-2x). Then, we have:
y1(x) = e^(4x), y2(x) = e^(-2x)W(y1,y2) = det([e^(4x), e^(-2x)], [4e^(4x), -2e^(-2x)]) = -6f(x) = 3e^(2x), g(x) = -2e^(-x)u1'(x) = -(y_h(x)g(x))/W(y1,y2) = -[(c1e^(4x) + c2e^(-2x))(-2e^(-x))]/(-6) = (1/5)e^(4x) - (3/10)e^(2x)u2'(x) = (y_h(x)f(x))/W(y1,y2) = [(c1e^(4x) + c2e^(-2x))(3e^(2x))]/(-6) = (2/5)e^(-x)Integrating u1'(x) and u2'(x) with respect to x, we get:
u1(x) = -(1/10)e^(4x) + (1/5)e^(2x)u2(x) = -(2/5)xe^(-x)So, the particular solution is y_p(x) = -(1/10)e^(4x) + (1/5)e^(2x) - (2/5)xe^(-x).
Find the general solution to the differential equation with the initial conditions
The general solution to the differential equation is y(x) = y_h(x) + y_p(x) = c1e^(4x) + c2e^(-2x) + -(1/10)e^(4x) + (1/5)e^(2x) - (2/5)xe^(-x).
Using the initial conditions y(0) = 1 and y'(0) = 0, we get:
y(0) = c1 + c2 - (1/10) + (1/5) = 1y'(x) = 4c1e^(4x) - 2c2e^(-2x) + (1/5)e^(2x) + (2/5)xe^(-x)y'(0) = 4c1 - 2c2 + (1/5) = 0Solving these equations gives c1 = 3/5 and c2 = 2/5. So, the final solution is:
y(x) = (1/10)e^(2x) + (3/5)e^(-x) - (1/10)e^(4x) - (2/5)xe^(-x)To learn more about differential equation, here
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Related Questions
a deck of cards has three red cards, three blue cards, three yellow cards, and three green cards. you shuffle the deck and draw exactly two cards. what is the number of ways you can draw two cards of the same color?
Answers
There are 12 combination ways to draw two cards of the same color from the shuffled deck, since there are 6 pairs of cards of the same color to choose from.
1. There are a total of 12 cards with 3 cards of each color (red, blue, yellow, and green).
2. To draw two cards of the same color, you must choose one of the six pairs of cards of the same color, which can be done in 6 ways.
3. Therefore, there are 6 x 1 = 6 possible ways to draw two cards of the same color from the shuffled deck.
If you draw three cards combination instead of two, there are still 6 ways to draw cards of the same color, since you must still choose one of the six pairs of cards of the same color. This can be done in 6 x 1 x 1 = 6 ways.
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Suppose 20% of ohio residents support the legalization of marijuana. If you randomly select n people and would like to use the normal approximation to answer questions, what does your sample size have to be, at minimum?.
Answers
The minimum sample size for the given event to have a normal approximation has to be 13 people.
For normal approximation,
np or n(1-p) should be greater than 10
where,
n = the sample size
p = probability for the occurrence of the event
Here, 20% of ohio residents support the legalization of marijuana
p = 20% or, 2/10
Hence,
1 - p = 1 - 2/10
= 8/10
Therefore, for normal approximation to be used
either n X 2/10 > 10 or, n X 8/10 > 10
Therefore,
n X 2 > 10 X 10 or, n X 8 > 10 X 10
hence,
n > 100/2 or, n > 100/8
n > 50 or, n > 12.5
Since we need to find the minimum no. of sample size to be required we will consider
n > 12.5
The sample size has to be a whole number hence,
n = 13
Therefore, the minimum sample size for the given event to have a normal approximation has to be 13 people.
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I"LL GIVE BRAINLY!!!
Natalee earned m dollars a day for 4 days. How many total dollars did she make?
m + 4
m/4
m - 4
4m
Answers
write the number 3.8 in the form a/b using integer
Answers
b) After allowing 16% discount on the marked price of a watch, 13% Value Added Tax (VAT) was levied on it. If the watch was sold for Rs 4,746, calculate the marked price of the watch.
Answers
Let's denote the marked price as "M."
Given:
Selling price after discount and VAT = Rs 4,746
Discount = 16%
VAT = 13%
Step 1: Calculating the selling price before VAT
Let's assume the selling price before VAT as "X."
X = Selling price after VAT / (1 + VAT rate)
X = Rs 4,746 / (1 + 0.13)
X = Rs 4,746 / 1.13
X ≈ Rs 4,200
Step 2: Calculating the marked price before discount
Let's assume the marked price before the discount as "Y."
Y = Selling price before VAT / (1 - Discount rate)
Y = Rs 4,200 / (1 - 0.16)
Y = Rs 4,200 / 0.84
Y ≈ Rs 5,000
Therefore, the marked price of the watch would be approximately Rs 5,000.
Please note that the actual marked price may have been rounded to the nearest value in the given calculation.
?
-22/35+14/15
pls hurry I need help.
answer as a reduced proper or improper fraction
Answers
Answer:
Step-by-step explanation:
32/105
If 10g of a radioactive substance are present initially and 9 yr later only 5.0g remain, how much of the substance, to the nearest tenth of a gram, will be present after 14 yr? After 14 yr, there will be ___g of the radioactive substance. (Do not round until the final answer. Then round to the nearest tenth as needed.)
Answers
after 14 years, there will be approximately 1.97g of the radioactive substance remaining.
To solve this problem, we can use the concept of exponential decay for radioactive substances. The decay of a radioactive substance can be modeled by the equation:
N(t) = N₀ * e^(-kt),
where N(t) is the amount of substance at time t, N₀ is the initial amount of substance, e is the base of the natural logarithm, k is the decay constant, and t is the time elapsed.
In this case, we are given that the initial amount N₀ is 10g and the amount after 9 years N(9) is 5g. We can use this information to find the value of k.
5 = 10 * e^(-9k).
Divide both sides of the equation by 10:
0.5 = e^(-9k).
Take the natural logarithm of both sides:
ln(0.5) = -9k.
Solve for k:
k = ln(0.5) / -9.
Now that we have the value of k, we can use it to find the amount of substance after 14 years, N(14):
N(14) = N₀ * e^(-kt).
N(14) = 10 * e^(-ln(0.5) / -9 * 14).
N(14) = 10 * e^(14 * ln(0.5) / 9).
N(14) ≈ 10 * 0.197 = 1.97.
Therefore, after 14 years, there will be approximately 1.97g of the radioactive substance remaining.
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combien de morceaux de 15cm peut on découper dans une ficelle de 1cm ?
Answers
Answer:
English translate for my help please??!
Step-by-step explanation:
Rafael brought 16 packs of cola to a party, and each pack had 6 cans. Maria brought 9 cans of juice. How many cans did they bring in all?
Answers
For this question, we will take the number of packs of cola which is 16, and multiple it by 6 which gives you 96 then you add the 9 cans that Maria brought which means your answer is 105
Hope this helps
Consider the scatter plot
What type of association does the scatter plot show?
Drag a word or phrase into the box to correctly complete the
sentence
The scatter plot shows
association
NEED done today
Answers
The given scatter plot shows a negative association.
Scatter Plot and Association RelationshipWhen a scatter plot shows an upward trend, that is, the trendline sloping upwards to the right, it shows a positive association.When the trendline slopes downwards from the left to the right, it shows a negative association between two variables.Therefore, the given scatter plot shows a negative association.
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Answer:
Negative
Step-by-step explanation:
need this answer quick due soon help
Answers
Answer:
2/9
Step-by-step explanation:
Let 0.2222..... be x
10x = 2.2222222......
=> 10x - x = 2.22222..... - 0.22222....
=> 9x = 2
=> x = 2/9
first sequence: 3 8 13 18 23 second sequence: -2 4 10 16 . find the TWO numbers that are in both number sequences.
Answers
\(a(n) = 3 + 5n\)
\(b(n) = - 2 + 6n\)
\(a(n) = b(n) \\ 3 + 5n = - 2 + 6n \\ 5 = n \\ n = 5 \: (6th \: term)\)
\(a(5) \: or \: b(5) = 3 + 5(5) = 28\)
Each year, roughly 10^6 computer programmers each make about $ 10^5.
How much money is this all together? Express your answer both as a power
of 10 and as a dollar amount.
Answers
HELP.................................................................
Answers
Answer:
A. \(\frac{1}{6^{2} }\)
Step-by-step explanation:
A. \(\frac{1}{6^{2} } = \frac{1}{36} = 0.02777\)
B. \(6^{-3} = 0.004629\)
C. \(\frac{1}{216} = 0.004629\)
D. \(\frac{1}{6^{3} } = \frac{1}{216} = 0.004629\)
Answer:
I think it is A because B or D is not it so I was left with A and C. So I am pretty sure that is A.
what two numbers would give a sum of 30, a difference of 24, product of 81 and a quotient of 9
Answers
Answer:
27 and 3
Step by Step explanation:
Just list the factors of 81 and find the two numbers
State which metric unit you would probably use to measure item.
Mass of a beach ball
Answers
The basic unit of mass of the ball in the metric system is the gram.
What is mass?The base units of length (distance), capacity (volume), and weight (mass) in the metric system are the meter, liter, and gram, respectively. We utilize units that are derived from metric units to measure smaller or greater quantities.
Mass is a physical body's total amount of matter. Inertia, or the body's resistance to acceleration when a net force is applied, is also measured by this term. The strength of an object's gravitational pull to other bodies is also influenced by its mass.
Therefore, the basic unit of mass of the ball in the metric system is the gram.
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Chris invests $20,000 into a savings account that accrues 5.2% interest compounded semiannually. What is the balance of Chris' account after 3 years?
Answers
The amount to be paid after 3 years is $23,329.97.
What is Compound Interest?In order to compute compound interest, multiply the principle of the original loan by the annual interest rate multiplied by the number of compound periods minus one. You will then be left with the principal amount of the loan plus compound interest.
Given:
P= 20,000
r= 5.2%
t= 3 years
r = R/100
r = 5.2/100
r = 0.052 rate per year,
Now, solve for A
A = P\((1 + r/n)^{nt\)
A = 20,000.00\((1 + 0.052/2)^{(2)(3)\)
A = 20,000.00\((1 + 0.026)^(6)\)
A = $23,329.97
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Solve the initial-value problem 2y" + 5y' - 3y = 0, y(0) = 3, y'(0) = 19.
Answers
In summary, we are given a second-order linear homogeneous differential equation, 2y" + 5y' - 3y = 0, along with initial conditions y(0) = 3 and y'(0) = 19. We need to find the solution to this initial-value problem.
To solve this initial-value problem, we can use the method of undetermined coefficients or the characteristic equation. Since the equation is linear and homogeneous, we can use the characteristic equation approach. We assume a solution of the form y(t) = e^(rt), where r is a constant to be determined. By substituting this form into the differential equation, we obtain the characteristic equation 2r^2 + 5r - 3 = 0.
Solving the characteristic equation, we find two distinct roots: r = 1/2 and r = -3/2. Therefore, the general solution to the differential equation is y(t) = c₁e^(1/2t) + c₂e^(-3/2t), where c₁ and c₂ are constants determined by the initial conditions. Plugging in the initial conditions y(0) = 3 and y'(0) = 19, we can set up a system of equations to solve for the constants. Finally, substituting the values of c₁ and c₂ back into the general solution, we obtain the specific solution to the initial-value problem.
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13, 6 7 What is the coefficient of mon in the expansion of (m-2n)¹3? The coefficient is (Simplify your answer.)
Answers
the coefficient of m is -6n when we expand (m - 2n)³.Therefore, the answer is -6.
To find the coefficient of m in significant figures the expansion of (m - 2n)³, the formula to use is as follows:
Here, a = m and b = -2n. Thus, the given expression can be written as Expanding the expression,
we get \($$(m - 2n)^3 = \binom{3}{0}m^3(-2n)^0 + \binom{3}{1}m^2(-2n)^1 + \binom{3}{2}m^1(-2n)^2 + \binom{3}{3}m^0(-2n)^3$$$$\Rightarrow (m - 2n)^3 = m^3 - 6m^2n + 12mn^2 - 8n^3$$\)
Thus, the coefficient of m is -6n when we expand (m - 2n)³.Therefore, the answer is -6.
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As a nurse, part of your daily duties is to mix medications in the proper proportions for your patients. For one of your regular patients, you always mix Medication A with Medication B in the same proportion. Last week, your patient's doctor indicated that you should mix 80 milligrams of Medication A with 48 milligrams of Medication B. However this week, the doctor said to only use 6 milligrams of Medication B. How many milligrams of Medication A should be mixed this week?
Answers
Answer:
Hey! I see you need some help, I gotchu.so heres what i got,
Medication A: 32 would be the main answer
Step-by-step explanation:
Week 1- even medication from A and medication B
Week2- Medication A: 80 milligrams Medication B: 48 milligrams
week 3- Medication A:? Medication B: 6
They seem to be reducing it by 42, so 48-42=6 (medication B)
So therefore, 80-42=32. would be for medication A
A=8; b= ____; c=10 what is the answer
Answers
Answer:
b=6 is your answer
g Suppose the time (minutes) an Instructor spends grading a Math 10 quiz is uniformly distributed in the interval (8, 25). Find the probability that the Instructor spends more than 16 minutes grading a math 10 quiz.
Answers
The probability that the instructor spends more than 16 minutes grading the quiz is:P(X > 16) = 1 - P(X ≤ 16) = 1 - 8 / 17 = 9 / 17. The probability that the instructor spends more than 16 minutes grading a Math 10 quiz is 9 / 17.
First of all, we need to calculate the probability that the instructor spends 16 minutes or less grading the quiz. After that, we can subtract that probability from 1 to obtain the probability that the instructor spends more than 16 minutes grading the quiz.Let X be the time (minutes) an instructor spends grading a Math 10 quiz. Since X is uniformly distributed in the interval (8, 25), we have:X ~ U(8, 25)
The probability density function (PDF) of X is given by:f(x) = 1 / (25 - 8) = 1 / 17, for 8 ≤ x ≤ 25.We can find the cumulative distribution function (CDF) of X by integrating the PDF as follows:F(x) = ∫f(t)dt = ∫1/17 dt = t / 17 + C, for 8 ≤ t ≤ xThe constant of integration C is determined by the condition that F(8) = 0, since P(X ≤ 8) = 0.
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If the level of significance of a hypothesis test is raised from 0.005 to 0.2, the probability of a type ii error will:__________
Answers
The probability of type II error will decrease if the level of significance of a hypothesis test is raised from 0.005 to 0.2.
What is a type II error?A type II error occurs when a false null hypothesis is not rejected or a true alternative hypothesis is mistakenly rejected.
It is denoted by 'β'. The power of the hypothesis is given by '1 - β'.
How the type II error is related to the significance level?The relation between type II error and the significance level(α):
The higher values of significance level make it easier to reject the null hypothesis. So, the probability of type II error decreases.The lower values of significance level make it fail to reject a false null hypothesis. So, the probability of type II error increases.Thus, if the significance level increases, the type II error decreases and vice-versa.From this, it is known that when the significance level of the given hypothesis test is raised from 0.005 to 0.2, the probability of type II error will decrease.
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can someone please help with this?
Answers
Answer:
each plan is 3/4 hours
Step-by-step explanation:
We can write equations for the total workout time on each day.
5A +3B = 6
7A +9B = 12
Subtracting the second equation from 3 times the first gives ...
3(5A +3B) -(7A +9B) = 3(6) -(12)
8A = 6 . . . . . . . . . simplify
A = 6/8 = 3/4
__
Using the simplified equation, we can substitute for 6 in the first equation.
5A +3B = 8A
3B = 3A . . . . . subtract 5A
B = A . . . . . . . divide by 3
B = 3/4
Each of Jessica's workout plans takes 3/4 hours.
If it takes 4/5 of an hour to paint 2/5 of a room how long does it take to painr one room
Answers
4/5 of 60 is 48 then gimes by 2/5
Joe is 3 times as old as Pepe In how many years will Joe be only twice as old as Pepe ? How old are they now? How old will they be? When will they be the same age?
Answers
Then, 3x = Jose's age now
Wally's age 5 years from now = x+5
Jose's age 5 years from now = 3x+5
(x+5) + (3x+5) = 42
4x + 10 = 42
4x = 32 x = 8
Wally is 8 years old and Jose is 24.
Is the graphed function linear?
Yes, because each input value corresponds to exactly one output value.
Yes, because the outputs increase as the inputs increase.
No, because the graph is not continuous.
No, because the curve indicates that the rate of change is not constant.
Answers
Answer:
The answer is D. No, because the curve indicates that the rate of change is not constant.
Step-by-step explanation:
There is not a constant rate of change which results in a curved line.
Answer:
It is the last one: No, because the curve indicates that the rate of change is not constant.
Step-by-step explanation:
You can tell in the picture that if you connected the points it would not create a straight line.
a boat travels at 3.8m/s east across a river 240 m wide with a current of 1.6 m/s south. how long does it take to get across?
Answers
Answer:
The time it takes to get across is 240m divided by 3.8 m/s, which is 63.2 seconds. In that time the current has moved boat downstream by: 1.6 m times 63.2 seconds, or 101.1 meters. Then to get the resultant velocity, you need to add the velocity vectors. Use pythagorean theorem: resultant^2 = 3.8^2 + 1.6^2. The resultant velocity is 4.1 m/s. Does this make sense??
mark brilliant pls
I need help on this giving brainlist ASAP
Answers
Answer:
no solution
Step-by-step explanation:
|x| = -10
Absolute value means that it is positive or 0
It cannot be negative
There is no solution
Step-by-step explanation:
There is no solution
|A|
can be only 0 or positive number
can't be -
from the rule of Mathematics
Work out the value of (312 ÷ 35) ÷ (32 × 3)
Answers
Answer:
\((312 \div 35) \div (32 \times 3) \\ \\ 8.914 \div 96 \\ \\ = 0.09\)