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Laplace transform of t2 sin(2t).- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
Laplace transform of t2 sin(2t).
a)
b)
c)
d)
|
Sanya Agarwal answered |
We know that,
Here, f(t) = sin(2t) => F(s) =
Hence, L(t2sin(2t)) =
Here, f(t) = sin(2t) => F(s) =
Hence, L(t2sin(2t)) =
The Laplace transform of the differential equation y" + ay' + by = f(t). Assume that y(0) = 5, y'(0) = 10, Y(s) and F(s) are the Laplace transforms of y(t) and f(t) respectively- a)s2Y(s) + 10s + 5 + a(sY(s) + 10) + bY(s) = F(s)
- b)s2Y(s) - 5s - 10 + a(sY(s) - 5) + bY(s) = F(s)
- c)s2Y(s) - 10s - 5+ a(sY(s) - 10) + bY(s) = F(s)
- d)s2Y(s) + 5s + 10+ a(sY(s) + 5) + bY(s) = F(s)
Correct answer is option 'B'. Can you explain this answer?
The Laplace transform of the differential equation y" + ay' + by = f(t). Assume that y(0) = 5, y'(0) = 10, Y(s) and F(s) are the Laplace transforms of y(t) and f(t) respectively
a)
s2Y(s) + 10s + 5 + a(sY(s) + 10) + bY(s) = F(s)
b)
s2Y(s) - 5s - 10 + a(sY(s) - 5) + bY(s) = F(s)
c)
s2Y(s) - 10s - 5+ a(sY(s) - 10) + bY(s) = F(s)
d)
s2Y(s) + 5s + 10+ a(sY(s) + 5) + bY(s) = F(s)
|
Dipanjan Nambiar answered |
Understanding the Differential Equation
The given differential equation is:
y'' + ay + by = f(t)
To solve it using the Laplace transform, we need to apply the transform to each term in the equation.
Applying Laplace Transforms
1. Transform of y'':
- The Laplace transform of y''(t) is s²Y(s) - sy(0) - y'(0).
2. Transform of y:
- The Laplace transform of y(t) is Y(s).
3. Transform of f(t):
- This is simply F(s).
Substituting Initial Conditions
Given initial conditions are:
- y(0) = 5
- y'(0) = 10
Plugging these values into the Laplace transform:
1. For y'':
- s²Y(s) - 5s - 10
2. For y:
- a(sY(s) - 5) (since y(0) = 5)
3. Combine Everything:
- The full equation becomes:
s²Y(s) - 5s - 10 + a(sY(s) - 5) + bY(s) = F(s)
Final Formulation
From the steps above, we can simplify:
- The correct equation is:
s²Y(s) - 5s - 10 + a(sY(s) - 5) + bY(s) = F(s)
This matches option b, confirming its correctness.
Conclusion
Therefore, the correct answer is option 'b', as it accurately reflects the application of the Laplace transform to the given differential equation with the specified initial conditions.
The given differential equation is:
y'' + ay + by = f(t)
To solve it using the Laplace transform, we need to apply the transform to each term in the equation.
Applying Laplace Transforms
1. Transform of y'':
- The Laplace transform of y''(t) is s²Y(s) - sy(0) - y'(0).
2. Transform of y:
- The Laplace transform of y(t) is Y(s).
3. Transform of f(t):
- This is simply F(s).
Substituting Initial Conditions
Given initial conditions are:
- y(0) = 5
- y'(0) = 10
Plugging these values into the Laplace transform:
1. For y'':
- s²Y(s) - 5s - 10
2. For y:
- a(sY(s) - 5) (since y(0) = 5)
3. Combine Everything:
- The full equation becomes:
s²Y(s) - 5s - 10 + a(sY(s) - 5) + bY(s) = F(s)
Final Formulation
From the steps above, we can simplify:
- The correct equation is:
s²Y(s) - 5s - 10 + a(sY(s) - 5) + bY(s) = F(s)
This matches option b, confirming its correctness.
Conclusion
Therefore, the correct answer is option 'b', as it accurately reflects the application of the Laplace transform to the given differential equation with the specified initial conditions.
Determine the time signal x(t) corresponding to given X (s) and choose correct option.
Q. 
- a)(e-2t + 2e-3t cos t + 2e-3t sin t)u(t)
- b)(e-2t + 2e-3t cos t - 6e-3t sin t)u(t)
- c)(e-2t + 2e-3t cos t - 2e-3t sin t)u(t)
- d)(9e-2t + 6e-3t cos t + 3e-3t sin t)u(t)
Correct answer is option 'B'. Can you explain this answer?
Determine the time signal x(t) corresponding to given X (s) and choose correct option.
Q.
Q.
a)
(e-2t + 2e-3t cos t + 2e-3t sin t)u(t)
b)
(e-2t + 2e-3t cos t - 6e-3t sin t)u(t)
c)
(e-2t + 2e-3t cos t - 2e-3t sin t)u(t)
d)
(9e-2t + 6e-3t cos t + 3e-3t sin t)u(t)
|
Saranya Mishra answered |
Determine the time signal x(t) corresponding to given X (s) and choose correct option.
Q. 
- a)(2e-2t + 2e-t sin 2t - 2e-t cos 2t )u(t)
- b)(2e-2t + 2e-t cos 2t - 2e-t sin 2t )u(t)
- c)(2e-2t + 2e-t cos 2t - e-t sin 2t )u(t)
- d)(2e-2t + 2e-t sin 2t - e-t cos 2t )u(t)
Correct answer is option 'C'. Can you explain this answer?
Determine the time signal x(t) corresponding to given X (s) and choose correct option.
Q.
Q.
a)
(2e-2t + 2e-t sin 2t - 2e-t cos 2t )u(t)
b)
(2e-2t + 2e-t cos 2t - 2e-t sin 2t )u(t)
c)
(2e-2t + 2e-t cos 2t - e-t sin 2t )u(t)
d)
(2e-2t + 2e-t sin 2t - e-t cos 2t )u(t)
|
Aman Datta answered |
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