Question

use the laplace transform to solve the given initial-value problem. y'' − 4y' 4y = t, y(0) = 0, y'(0) = 1

Answer 1

To solve the given initial-value problem using Laplace transforms, take the Laplace transform of the differential equation, decompose Y(s) into partial fractions, and then take the inverse Laplace transform to find the solution y(t).

Step-by-step explanation:

To solve the given initial value problem using Laplace transforms, we will first take the Laplace transform of the given differential equation. Applying the Laplace transform to the equation y'' - 4y' + 4y = t, we get (s^2 - 4s + 4)Y(s) = 1/s^2. Simplifying this equation, we find Y(s) = 1/(s^2)(s-2)(s-2).

We can now use partial fraction decomposition to express Y(s) in terms of simpler fractions. Decomposing Y(s) into partial fractions, we get Y(s) = A/s^2 + B/(s-2) + C/(s-2)^2. By equating the numerators and finding the values of A, B, and C, we obtain A = -1/4, B = 1/2, and C = -1/4.

Taking the inverse Laplace transform of Y(s), we find the solution y(t) = -1/4 + 1/2e^(2t) - 1/4te^(2t).

Answer 2

To solve the initial-value problem using the Laplace transform, we apply the transform to the differential equation, utilize the initial conditions to solve for Y(s), potentially use partial fraction decomposition, and finally find y(t) by taking the inverse Laplace transform.

Step-by-step explanation:

The student has asked us to use the Laplace transform to solve the initial-value differential equation y'' − 4y' + 4y = t, with the initial conditions y(0) = 0 and y'(0) = 1. For the Laplace transform of the second derivative, we'll use the formula L{y''} = s^2Y(s) - sy(0) - y'(0), and similarly for the first derivative. We then get an algebraic equation in terms of Y(s) that we can solve for Y(s). Finally, we take the inverse Laplace transform to find y(t).

Step-by-Step Solution:

1. Take the Laplace transform of both sides of the equation:
2. Plug in the initial conditions and solve the resulting algebraic equation for Y(s).
3. Use partial fraction decomposition if necessary.
4. Take the inverse Laplace transform of Y(s) to get y(t).