Final answer:
The question involves solving a linear homogeneous differential equation to find the motion characteristics of an object described by position, velocity, and acceleration as functions of time. This requires applying initial conditions to determine the constants and evaluating the motion at specific times.
Step-by-step explanation:
The provided information relates to solving differential equations and analyzing the motion of an object in terms of its position, velocity, and acceleration as functions of time. The system described, dx/dt = [[ -11 18; -3 4 ]] x, is a first-order linear homogeneous differential equation with constant coefficients. Solving this involves finding the eigenvalues and eigenvectors of the matrix, which will help in constructing the general solution of the system. The exercise highlighted involves applying initial conditions to find specific solutions and determining the constants of integration.
In various contexts discussed, the position x(t), velocity v(t), and acceleration a(t) functions are derived and evaluated at specific times to find the motion characteristics of an object. Different approaches to solving for time, t, when given position and velocity are also outlined, utilizing kinematic equations and solving quadratic equations.