Question

Show how to solve this differential equation using laplace

d y/d t-4 y=-7 H(t-2), subject to the boundary condition y(0)=-4

Answer 1

In solving the given differential equation with Laplace transforms, take the Laplace of both sides, substitute boundary conditions, solve for Y(s), and finally find the inverse Laplace to get the solution y(t).

Step-by-step explanation:

To solve the differential equation dy/dt - 4y = -7H(t-2), subject to the boundary condition y(0) = -4, using the Laplace transform, follow these steps:

1. Take the Laplace transform of both sides of the differential equation. The Laplace transform of dy/dt is sY(s) - y(0), and for y it is just Y(s).
2. Substitute the initial condition and the Laplace transform of the Heaviside function H(t-2) to get an algebraic equation in s.
3. Solve the algebraic equation for Y(s) by isolating Y(s) on one side.
4. Finally, find the inverse Laplace transform of Y(s) to get the solution y(t) in the time domain.

Using the steps above, we find:

1. The Laplace transform of the equation is (sY(s) - y(0)) - 4Y(s) = -7/s for t > 2.
2. Substituting the boundary condition y(0) = -4 gives (sY(s) + 4) - 4Y(s) = -7/s, which simplifies to sY(s) - 4Y(s) = -7/s - 4.
3. Rearranging, we get Y(s)(s - 4) = -7/s - 4. Solve for Y(s) to get Y(s) = (-7/s - 4)/(s - 4).
4. The inverse Laplace transform of Y(s) yields the solution y(t).