Use the Laplace transform to solve the given initial value problem. y' + 6y = e^4t ; y(0)=2 ...?
Answers
sY(s) - y(0) + 6Y(s) = 1/(s-4)
(s+6)Y(s) - 2 = 1/(s-4)
Y(s) = 2/(s+6) + 1/(s-4)(s+6), by partial fraction decomposition
Y(s) = 2/(s+6) + 1/10 * (1/(s-4) + 1/(s+6))
Y(s) = 0.1/(s-4) + 2.1/(s+6)
Performing inverse laplace transform,
y(t) = 0.1e^4t + 2.1e^(-6t)
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Final answer:
The Laplace transform method is applied to solve the differential equation y' + 6y = e^4t with the initial condition y(0)=2. After transforming, simplifying, and solving for Y(s), we use inverse Laplace transform to find the solution y(t) in the time domain.
Explanation:
Laplace transform is a powerful tool in the field of mathematics used for solving differential equations. To solve the given initial value problem y' + 6y = e^4t ; y(0)=2, we can start by taking the Laplace transform of both sides of the equation.
The Laplace transform of y' is sY(s) - y(0) and the Laplace transform of y is Y(s). Therefore, the Laplace transform of y' + 6y gives sY(s) - y(0) + 6Y(s). Given that y(0)=2, this simplifies to sY(s) + 6Y(s) - 2.
On the right-hand side, the Laplace transform of e^4t is 1/(s-4). Thus, we have the equation sY(s) + 6Y(s) - 2 = 1/(s-4).
By solving for Y(s), we can find the inverse Laplace transform to get the solution y(t) in the time domain.
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Related Questions
Six less than a number is less than the quotient of the number and three
HELP!!! is the answer for b correct? 10 points and marked brainliest
State an equation you could use to find the value of x. Then find the value of x in simplest form.
HELP ITS DUE IN 15 mins!!!!! and i need help:(
Answers
The expression of 4e, that has the coefficient is 4
What are the Quotients?
Quotients are the number that is obtained by dividing one number by another number.
The Numbers constants, variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbol; which can also be used to indicate the logical syntax's order of operations and other features.
Dividend ÷ Divisor = Quotients
A coefficient is a number placed B4 another quantity for this case; that is the unknown(e).
The expression for the given situation is;
4e
Therefore , the coefficient is 4
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Answer:
4e
A coefficient is a number placed b4 another quantity for this case that is the unknown(e).
Therefore the coefficient is 4
Step-by-step explanation:
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B
The ratio surface area of toy A : surface area of toy B is 7:9
The volume of toy B is 20 cm
I
Work out the volume of toy A.
Give your answer correct to 3 significant figures.
Answers
The volume of toy A will be equal to 13.7 cubic centimetres.
What are ratios and proportions?
An ordered pair of numbers a and b, represented as a / b, is a ratio if b is not equal to 0. A proportion is an equation that sets two ratios at the same value.
The numerical relationship between two values demonstrates how frequently one value contains or is contained within another.
Given that the ratio surface area of toy A: to the surface area of toy B is 7:9. The volume of toy B is 20 cm
The computation of the volume of toy A is shown below:
7 × k²=9
k²= 9 ÷ 7
k=√(9 ÷ 7)
k = 1.133893419
Now
The volume of toy A = 20 ÷ 1.133893419³
The volume of toy A = 13.7 cubic centimetres
Therefore, the volume of toy A will be equal to 13.7 cubic centimetres.
The missing figure is attached with the answer below.
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Answer:
13.7
Step-by-step explanation:
7*k²=9
k²=9/7
k=√9/7
k=1.133893419
20/1.133893419³=13.7
Answers
Answers
The answer is C. 7.1 inches
Answers
'of' in mathematics means multiplication
percent means over 100
32 = 40% * x
32 = 40/100 * x
32 = 4/10 * x
Multiply 10/4 on both sides, so we can isolate 'x'
32 * 10/4 = x
x = 8 * 10
x = 80
32 is 40 percent of 80.
What is the value of x when f (x) = 4?
Answers
Answer:f(4)= 12
Step-by-step explanation:
Treat f(x) as y, y=2x + 4, y=12
Answer:
x = 0
Step-by-step explanation:
Given f(x) = x² + 4 and f(x) = 4, then equating
x² + 4 = 4 ( subtract 4 from both sides )
x² = 0 , thus
x = 0