Question

Using Laplace Transforms determine the solution y(t), to the following initial value problem dy/dt−4y=−7H(t−2), subject to the boundary condition y(0)=−4.

Answer 1

To solve the given initial value problem using Laplace Transforms, apply the Laplace transform to the equation and use the Laplace transform of the Heaviside step function. Then, solve for the Laplace transform of the unknown function and take the inverse Laplace transform to find the solution in the time domain.

Step-by-step explanation:

The given problem is to find the solution y(t) of the initial value problem using Laplace Transforms. The initial value problem is dy/dt - 4y = -7H(t-2) and the boundary condition is y(0) = -4. To solve this, we can apply the Laplace transform to both sides of the equation and use the Laplace transform of the Heaviside step function to simplify the expression.

Applying the Laplace transform, we get sY(s) - 4Y(s) = -7e^(-2s)/(s+7), where Y(s) represents the Laplace transform of y(t). Solving for Y(s), we find Y(s) = -7e^(-2s)/[(s+4)(s+7)].

Now, we take the inverse Laplace transform of Y(s) to find the solution y(t) in the time domain. The inverse Laplace transform of Y(s) is y(t) = -7e^(-4t) +7e^(-7t).