find the function y(t) that satisfies the differential equation dy dt−2ty=12t2et2 and the condition y(0)=−3.
Answers
We can use the initial condition y(0) = -3 to solve for the constant C. Plugging in t = 0 and y = -3 into the above equation.
To solve the given differential equation, we can use the method of integrating factor.
The given differential equation is: dy/dt - 2ty = \(12t^2\times e^(t^2)\)
First, we identify the integrating factor (IF), which is given by the exponential of the integral of the coefficient of y. In this case, the coefficient of y is -2t, so we integrate -2t with respect to t:
IF = e^∫(-2t)dt
IF = e^(-t^2)
Next, we multiply the entire differential equation by the integrating factor:
\(e^(-t^2)\times (dy/dt - 2ty) = e^(-t^2) \times 12t^2\times e^(t^2)\)
This simplifies to:
\(e^(-t^2) \times dy/dt - 2ty \times e^(-t^2) = 12t^2\)
Now, we can integrate both sides of the equation with respect to t. Integrating the left-hand side (LHS) requires integration by parts, while the right-hand side (RHS) can be integrated directly:
∫(e^(-\(t^2\)) \(\times\) dy/dt) dt - ∫(2ty \(\times\) e^(-\(t^2\))) dt = ∫(12\(t^2\)) dt
Using integration by parts on the LHS with u = e^(-\(t^2\)) and dv = dy/dt dt, we get:
e^(-\(t^2\)) \(\times\)y - ∫(e^(-\(t^2\)) \(\times\) y') dt - ∫(2ty \(\times\) e^(-\(t^2\))) dt = 12\(t^3\) / 3 + C
where C is the constant of integration.
Rearranging the equation and simplifying, we get:
e^(-\(t^2\)) \(\times\) y - ∫(e^(-\(t^2\)) \(\times\) y') dt - 2∫(ty \(\times\) e^(-\(t^2\))) dt = 4\(t^3\) + C
Now, we can integrate the remaining integrals. For the second term on the left-hand side, we use u = t and dv = y \(\times\) e^(-\(t^2\)) dt, and for the third term on the left-hand side, we use u = y and dv = t \(\times\) e^(-\(t^2\)) dt. This gives us:
e^(-\(t^2\)) \(\times\) y - (y \(\times\) e^(-\(t^2\))) - (e^(-\(t^2\))\(\times\) t \(\times\) y) = 4\(t^3\) + C
Combining like terms and multiplying both sides by e^(t^2) to eliminate the exponential factor, we get:
y - y \(\times\) e^(-\(t^2\)) - t \(\times\) y = 4\(t^3\) \(\times\) e^(\(t^2\)) + C \(\times\) e^(\(t^2\))
Factoring out y on the left-hand side and simplifying, we get:
y \(\times\) (1 - e^(-\(t^2\)) - t) = 4t^3 \(\times\)e^(\(t^2\)) + C \(\times\)e^(\(t^2\))
Finally, dividing both sides by (1 - e^(-\(t^2\)) - t) to isolate y, we get:
y(t) = (4\(t^3\) \(\times\) e^(\(t^2\)) + C \(\times\) e^(\(t^2\))) / (1 - e^(-\(t^2\)) - t)
Now, we can use the initial condition y(0) = -3 to solve for the constant C. Plugging in t = 0 and y = -3 into the above equation.
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Related Questions
Your old high school pal Mike Errington wants to upgrade an old 1976 vintage room air conditioner that is believed to operate at an EER of 7. He is considering a room air conditioner with an EER of 13. He wants to know what percentage of electricity consumption would be reduced. Can you help him find it (answer must be in a percentage)
Answers
If Mike upgrades from an air conditioner with an EER of 7 to one with an EER of 13, he can calculate the percentage reduction in electricity consumption as follows:
Let's say the old air conditioner has a cooling capacity of C1 BTUs and a power input of P1 watts. The new air conditioner has a cooling capacity of C2 BTUs and a power input of P2 watts. Since the cooling capacity is the same for both air conditioners, we have:
C1 = C2
The power input for each air conditioner is:
P1 = C1 / 7 (since the EER is 7)
P2 = C2 / 13 (since the EER is 13)
Substituting C1 = C2, we get:
P1 = C2 / 7
P2 = C2 / 13
The percentage reduction in electricity consumption is:
(P1 - P2) / P1 x 100%
Substituting the expressions for P1 and P2, we get:
[(C2 / 7) - (C2 / 13)] / (C2 / 7) x 100%
Simplifying, we get:
(6 / 91) x 100% ≈ 6.59%
Therefore, if Mike upgrades to an air conditioner with an EER of 13, he can expect to reduce his electricity consumption by about 6.59%.
(Present value of an ordinary annuity) What is the present value of $2.500 per year for 10 years discounted back to the present at 7 percent? The present value of $2500 per year for 10 years discounted back to the present at 7 percent is : (Round to the nearest cent)
Answers
The present value of $2,500 per year for 10 years discounted back to the present at 7 percent is $17,462.03.
To calculate the present value of an ordinary annuity, we can use the formula:
PV = A * [1 - (1 + r)^(-n)] / r,
where PV is the present value, A is the annual payment, r is the discount rate per period, and n is the number of periods.
In this case, the annual payment is $2,500, the discount rate is 7 percent (or 0.07 as a decimal), and the number of periods is 10 years. Plugging in these values into the formula, we can calculate the present value:
PV = $2,500 * [1 - (1 + 0.07)^(-10)] / 0.07 ≈ $17,462.03.
Therefore, the present value of $2,500 per year for 10 years discounted back to the present at 7 percent is approximately $17,462.03. This represents the amount of money needed in the present to be equivalent to receiving $2,500 per year for 10 years with a 7 percent discount rate.
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The pie chart shows the sports played by 30 boys. How many boys play Rugby?
Answers
Answer:
10
Step-by-step explanation:
30 boys
1/2 play football
30/2 = 15
15 boys left
2/3 play rugby
15 x 2/3 = 10
Which is the best definition of a conic section ? cone and plane cone and line cone and point cone and circle
Answers
The correct option is cone and plane. A conic section is the intersection of a plane and a cone. The best definition of a conic section is "cone and plane".
The definition of conic section is a conic section is a shape that is formed when a right circular cone intersects a plane. Depending on the angle of the plane concerning the center line of the cone, the conic sections are classified into four types. They are a circle, parabola, ellipse, and hyperbola. Each type of conic section has a unique shape and properties.
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Write the slope-intercept form of the equation of each line.3) 6x - 8y = 404) -10 + 3x = -5y
Answers
Given data:
The expression for the equation of the line is 6x-8y=40.
The given equation can be written as,
6x-8y=40
8y=6x-40
y=6x/8-40/8
=3x/4-5
Thus, the slope intercept form is y=3x/4-5.
Part II: Writing & Solving Equations
1.
Wayne
paid
a total of $685
for his
accommodation
in Bali, Indonesia. Wayne
will
be spending 5
nights
at the hotel. The cost per night is $125
US Dollars
Answers
The equation representing the total accommodation cost is 110x = 685.
Therefore, Wayne would have to stay 6 nights if the cost per night is $110.
Given that Wayne paid a total of $685 for his accommodation in Bali, Indonesia. He will be spending 5 nights at the hotel. The cost per night is $125 US Dollars. To find the number of nights Wayne would have to stay if the cost per night was $110, Let us assume Wayne would have to stay for x number of nights if the cost per night is $110.The total cost of accommodation would be 110x, and the equation to represent the total cost of accommodation is 110x = 685
To find x, Dividing both sides by 110;110x/110 = 685/110x = 6.23 ≈ 6
Therefore, Wayne would have to stay 6 nights if the cost per night is $110.To check, The total cost of accommodation for 6 nights would be $110 × 6 = $660.
This is less than $685, so this is the correct answer. To write and solve the equation based on the above statement; Let the number of nights Wayne will stay at the hotel if the cost per night is $110 be represented as x.
The equation representing the total accommodation cost is 110x = 685. Now, to find the value of x;
Divide both sides by 110.110x/110 = 685/110x = 6.23 ≈ 6
Therefore, Wayne would have to stay 6 nights if the cost per night is $110.
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if p || , m<7 = 131°, and m<16 = 88°, find the measure of the missing angle m<8= ?
Answers
The angle 7 is given as, m<7 = 131°.
The sum of angle 7 and angle 8 is 180°.
\(m\angle7+m\angle8=180^{\circ}\)Substitute the m<7 = 131°.
\(171^{\circ}+m\angle8=180^{\circ}\)\(m\angle8=180^{\circ}-171^{\circ}\)\(m\angle8=49^{\circ}\)Thus the value of angle 8 is,
\(m\angle8=49^{\circ}\)What’s the answer guys .
Answers
Answer:
Hello! Your answer would be, D)
Step-by-step explanation:
Hope I helped! Ask me anything if you have more questions! Brainiest plz Have a nice mourning! Hope you make an 100%! -Amelia♥
HELP find the values of x (see image)
Answers
Answer:
4) 10
5) 15
6) 20
7) 4
8) 27
equations used to solve:
4) 4x+50=90
5) (2x+60)+6x=180
6) 3x+30=90
7) (8x+120)+7x=180
8) 99+3x=180
Reese had $42 in his pocket orl Sunday night. He spent $6.50 every day for lunch.
Which of the following equations best describes the relationship between d, the
number of days since Sunday, and m, the amount of money remaining?
Answers
The required equation is m = 42 - 6.50d which is the correct answer would be an option (C).
What is the Linear equation?A linear equation is defined as an equation in which the highest power of the variable is always one.
We have been given that Reese had $42 in his pocket on Sunday night. He spent $6.50 every day for lunch.
Let d represents the number of days since Sunday, and m represents the amount of money remaining
Reese has 42 total and it subtracts by 6.50$ every day
According to the given question,
The linear equation would be as
m = 42 - 6.50d
The above model equation is the relationship between d, the number of days since Sunday, and m, the amount of money remaining
Therefore, the required equation is m = 42 - 6.50d.
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A magician charges parties a $30 fee to cover travel and miscellaneous expenses plus $19.99 per hour. What an equation for that?
Answers
Answer:
30 + 19.99 per hour
Step-by-step explanation:
ADD THEM IN A EQUATION.
SO SIMPLE TO DO THAT!
Answer:
Step-by-step explanation:30+19.99x
Charle wants to order lunch for his friends. Hell order 6 sandwiches and $2 kids meals for his little brother. Charlie has 32$ how many can he spend on each sandwich if they are all the same price
Answers
Given:
Charle order 6 sandwiches and $2 kids meals.
Total amount he has = $32
Price of each sandwich is same.
To find:
The amount spend on each sandwich.
Solution:
Let x be the price of each sandwich.
Price of 6 sandwiches = 6x
Price of Kid meals = $2
Total price = \(6x+2\)
Total amount he has = $32
Now,
\(6x+2=32\)
\(6x=32-2\)
\(6x=30\)
Divide both sides by 6.
\(x=5\)
Therefore, the amount spend on each sandwich is $5.
Describe the solution set for the equation below.
−29+5x=5(x−5)
One Solution
One Solution
No Solution
No Solution
Infinitely Many Solutions
Answers
Please explain what are the advantages and disadvantages of
Opt-in go?
Answers
Advantages:
- It provides a more concise and explicit way to write code, reducing the likelihood of errors and making debugging simpler.
- It enables developers to write code focused on business logic, rather that the specifics of low-level language features.
- It is designed to take advantage of modern hardware, such as multiple cores and parallel processing, allowing for efficient and scalable code.
- The static typing system makes it easier to detect errors at compile-time, saving time in testing and debugging.
Disadvantages:
- The learning curve for the language can be steep, requiring a higher level of mastery to become fully productive, which can result in a delay in getting started on a project.
- As a relatively new language, some features may not yet be fully developed or may be missing entirely, making it harder to find resources and assistance.
- The developer community for Opt-in Go is not as large as some other programming languages, making it more difficult to find assistance and resources.
Andy is flying an airplane that is 12,000 feet in the air.
Answers
Answer:
D 21.80°
Step-by-step explanation:
Let x = measure of the angle
Tan x = 12000/30000 = .4
Arctan .4 = 21.80°
A basil plant grows about half an inch per week. When Finnegan buys the plant, it is 7
inches tall.
a. How tall is the basil plant after 1 week?
Answers
Answer:7 and a half
Step-by-step explanation:
Simple enough, if it seven inches and grows half an inch a week than after a week it grows half an inch, so its just 7 + a half, your awnser is 7 and a half
The temperature was -4°F this morning. If the temperature dropped 7°F, what is the temperature now?
Answers
Answer:
-11°F
Step-by-step explanation:
The temperature was -4° in the morning. The temperature dropped 7° later.
Subtract 7 from -4.
\(-4-7=\boxed{-11}\)
The temperature should be -11°F.
Hope this helps.
The temperature was -4°F this morning. If the temperature dropped 7°F then the temperature now should be -11° fahrenheit.
How are Kelvin, Celsius, and Fahrenheit related?We have got an equation that can relate these three units of measurement of temperature, as given below:
\(\dfrac{C}{5} = \dfrac{F - 32}{9} = \dfrac{K - 273}{5}\)
where C represents the measurement of a fixed temperature in celsius, F represents the measurement of that same intensity temperature in Fahrenheit, and also K represents the measurement of equally intense temperature in kelvin.
Given that the temperature was -4° in the morning. And the temperature dropped 7° later.
We have to Subtract 7 from -4.
-4 - 7 = 11
Thus, The temperature now should be -11°F.
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650 of those surveyed said their favorite color was blue. if 65% of the students surveyed said their favorite color was blue, how many students were surveyed in total?
Answers
Answer:
let X be the value of the total students
\( \frac{65}{100} \times x = 650 \\ \\ \frac{65x}{100 } = 650 \\ if \: w \:e \: crossmultipl \\ 65x = 65000 \\ x = 65000 \div 65 \\ x = 1000\)
Amare runs 1.10 miles in 2.3 what is his speed in miles per minute show your work.
Answers
Answer:
0.48 of a mile in 1 minute
Step-by-step explanation:
If Amare runs 1.10 miles in 2.3 minutes, you need to find how much he runs in 1 minute. 2.3 divided by what = 1? 2.3. Divide 1.10 by 2.3 to get 0.47826. You can round this to .48 or even .5, depending on the question. So, Amare runs .48 of a mile in 1 minute.
use geometry or symmetry, or both, to evaluate the double integral. 4ax3 8by3 a2 − x2 da, d = [−a, a] ⨯ [−b, b] d
Answers
By utilizing the principles of geometry and symmetry, we can evaluate the given double integral of the function 4a\(x^3\) + 8b\(y^3\)(\(a^2\) − \(x^2\))over the region defined by the rectangle D = [−a, a] ⨯ [−b, b].
To evaluate the double integral ∬D (4a\(x^3\) + 8b\(y^3\))(\(a^2\)- \(x^2\)) da, where D = [-a, a] × [-b, b], we can utilize symmetry to simplify the calculation.
First, let's focus on the term (\(a^2\) - \(x^{2}\)). This expression is symmetric with respect to x, meaning that the value of the integral over the interval [-a, a] will be twice the value of the integral over the interval [0, a].
Using this symmetry, we can rewrite the integral as follows:
∬D (4a\(x^3\)+ 8b\(y^3\))(\(a^2\) - \(x^2\)) da = 2∫₀ᵃ ∫₋₋bᵇ (4a\(x^3\) + 8b\(y^3\))(\(a^2\) -\(x^{2}\)) dy dx
Now let's evaluate the inner integral with respect to y:
∫₋₋bᵇ (4a\(x^3\)+ 8b\(y^3\))(\(a^2\) - \(x^{2}\)) dy
= (4a\(x^3\) + 8b\(y^3\))(\(a^2\) - \(x^{2}\))y | from -b to b
= (4a\(x^3\) + 8b(\(a^2\) - \(x^{2}\))\(b^3\)) - (4a\(x^3\) + 8b(\(a^2\) - \(x^{2}\))\((-b)^3\))
= 8\(b^4\)(\(a^2\) - \(x^{2}\))
Now we can substitute this result back into the double integral:
2∫₀ᵃ ∫₋₋bᵇ (4a\(x^3\) + 8b\(y^3\))(\(a^2\) - \(x^{2}\)) dy dx
= 2∫₀ᵃ 8\(b^4\)(\(a^2\) - \(x^{2}\)) dx
= 16\(b^4\) ∫₀ᵃ (\(a^2\) - \(x^{2}\)) dx
Now let's evaluate the remaining integral with respect to x:
16\(b^4\) ∫₀ᵃ (\(a^2\) - \(x^{2}\)) dx
= 16\(b^4\) [\(a^2\)x - (\(x^3\)/3)] | from 0 to a
= 16\(b^4\) (\(a^3\) - (\(a^4\)/3) - 0)
Simplifying further:
= 16\(b^4\) (\(a^3\) - (\(a^4\)/3))
= 16\(b^4\) (3\(a^3\) -\(a^4\))/3
Therefore, the value of the double integral ∬D (4a\(x^3\) + 8b\(y^3\))(\(a^2\) - \(y^3\)) da, over the region D = [-a, a] × [-b, b], is equal to 16\(b^4\) (3\(a^3\) - \(a^4\))/3.
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A family reunion has 184 people together at a local hotel. They travel in groups of 13 from the airport by van. How many vans will be needed to take the people to the hotel?
Answers
184 people
groups of 13
to find:How many vans needed to take the people to the hotel.
solution:\( \frac{184}{13} \)
\( = 14.1538461538\)
\(nearest \: whole \: number = 14\)
therefore, they'll need 14 vans to take the people to the hotel.
answer= option B
Rewrite in simplest term: 9(3s-3t)-2t-10(8t+s)
Answers
Answer:
17s-109t
Explanation:
Given the expression
\(9(3s-3t)-2t-10(8t+s)\)First, we open the brackets.
\(=27s-27t-2t-80t-10s\)Next, we collect terms with s together (do same for t).
\(=27s-10s-27t-2t-80t\)We then simplify to get:
\(=17s-109t\)What i an equation of the line that pae through the point (2, 3)(2,3) and (2, 7)(2,7)?
Answers
The equation of the line that passes through the points (2, 3) and (2, 7) is
= (y2 - y1)/(x2 - x1)
How do you find the equation for a line that passes through a point?These are the two approaches for determining a line's equation given a point and its slope: Plug the slope and the (x, y) point values into y = mx + b, then solve for b using the substitution technique. Create the equation y = mx + b using the m provided in the problem and the b that was just solved for.If the line crosses the point, the equation for the line's coordinates is solved at those points. We should obtain a valid expression if the x and y coordinates are entered into the line's equation.Only one straight line passes through two distinct points.
Given data :
The equation of the line that passes through the points (2, 3) and (2, 7) is
= (y2 - y1)/(x2 - x1)
= (7 - 3) / (2 - 2)
= 4 / 0 = 0
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pls help
Enter the equation for the graph.
Answers
Answer:
\(y=[1]cos([\frac{2\pi }{3}]x)\)
Step-by-step explanation:
Looking at the graph, we can see the domain to be from (0 , 2π).
Now we have to find one period that corresponds to cos(x).
The half-period of cos(x) for this graph appears to be pi/3 and adding another pi/3 gets us 2pi/3 to be our cosine period.
b = 2pi/3
a is the same range as cos(x). Range: (0,0)
y = [a] * cos ([b]*x)
y = [1] * cos([2pi/3]x)
3. A merchant sold a pen for $6.90, thereby making a profit of 15% on the cost to her. Calculate:
a. the cost price of the pen to the merchant to the nearest cent.
b. the selling price the merchant should request in order to make a 25% profit instead.
Answers
Answer:
A.$6.44
B.$8.05
Step-by-step explanation:
A.$6.90/15%=$0.46-$6.90=$6.44
B.$6.44*25%=$1.61+$6.44=$8.05
Answer:
answer for 6.90/15%= 6
answer for 25%= 7.5
Step-by-step explanation: These are answer is all correct there are no mistakes
x intercept of y = x-7
Answers
The x-intercept of the linear equation is (7, 0).
How to find the x-intercept of the linear function?Here we have the following linear equation:
y = x - 7
We want to find the x-intercept, this is the point where the line intercepts the x-axis, it will happen when we take the value of y = 0, then we need to solve.
0 = x - 7
Solving that, we get:
7 = x
Then we conclude that the x-intercept of the linear equation is (7, 0).
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Write a linear function f with f(5)=-1 and f(0)=-5
Answers
Answer:
Step-by-step explanation:
5. Which two triangles in problems 1-4 are similar? Explain.
Help please
Answers
Answer:
Triangle 1(ABC) and Triangle 3(XYZ) are similar since all their angles are equal
In Traingle XYZ m of angle Y is 85 degree..
Hope it helps u...
What is The geometric mean of 20 and 7
Answers
Answer:
Step-by-step explanation:
Formula
Geometric Mean = √(x1*x2)^(1/2)
This is the formula for 2 numbers. You multiply the two givens together and take the square root of the 2 numbers multiplied together. If there were 3 numbers, you would take the cube root.
Givens
x1 = 7
x2 = 20 Put the givens in the formula
Solution
Geometric mean = (7 * 20)^(1/2)
Geometric Mean = (140)^(1/2)
Geometric Mean = 11.83 Rounded.
a friend comes up to you and offers you a free ticket to a dodgers game that night, and you decide to attend the game. the game takes five hours and costs you $25 for transportation. if you had not attended the game, you would have worked at your part-time job for $12 an hour. what is the cost to you of attending the game?
Answers
The cost to you of attending the game instead of going to work is 85$
What are algebraic operations?They are the set of numbers and symbols that are related by the different mathematical operation signs such as addition, subtraction, multiplication, division among others.
To solve this problem we must perform the following algebraic operations with the given information
Information about the problem:
Transportation cost = 25$Hours paid for work = 12$ per hourTime of the game = 5 hCalculating how much will cost attending the game:
Total cost = (time of the game*Hours paid for work) + Transportation cost
Total cost = (12*5) + 25
Total cost = 85$
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