Question

5. Find the analytical solution to the initial value problem

dt
2

d
2
z

+9z=18H(t−3)cos(2t−6), subject to z=−1 and
z
˙
=4, when t=1. [4 marks]

Answer 1

The analytical solution to the differential equation with a Heaviside function requires using Laplace transforms and initial conditions to find the function z(t) in the time domain.

Step-by-step explanation:

The question asks to find the analytical solution to the initial value problem involving a second-order differential equation with a Heaviside function H(t-3) and a trigonometric forcing term, as well as the application of initial conditions z(1) = -1 and z'(1) = 4. This problem is within the scope of ordinary differential equations, a topic frequently encountered in college-level mathematics courses.

To tackle this problem, one would typically employ methods such as Laplace transforms to handle the Heaviside function and the given initial conditions. After finding the Laplace transform of the differential equation, solve for the transformed variable, and then take the inverse Laplace transform to find the solution z(t) in the time domain. The given initial conditions are used to solve for the constants that arise during the process.