Question

Use Laplace transforms to solve the following initial value problems.

d²y/dt² + 4dy/dt + 13y = u₅(t) , y(0) = 0, y′(0) = 1. Find limₜ→[infinity]ᵧ(t).

Answer 1

The student is asked to solve a differential equation using Laplace transforms and to find the steady-state solution as time approaches infinity.

Step-by-step explanation:

The question involves solving an initial value problem (IVP) with a differential equation using Laplace transforms. Here, the differential equation is d²y/dt² + 4dy/dt + 13y = u₅(t), with the initial conditions y(0) = 0 and y′(0) = 1. To solve this, the Laplace transform of the differential equation is taken, which converts it into an algebraic equation in the Laplace domain.

The Laplace transform of the derivative terms and the unit step function u₅(t) must be accounted for. After applying the initial conditions and simplifying, we can algebraically solve for Y(s), the Laplace transform of y(t). We then find the inverse Laplace transform to determine y(t). Finally, we want to find the limit of y(t) as t approaches infinity, which represents the steady-state solution of the system.