Final answer:
The question to find the solution to an initial value problem and describe its long-term behavior cannot be answered due to discrepancies in the information provided. A clarification of the exact differential equations and initial conditions is required to solve the problem correctly.
Step-by-step explanation:
The student is asking to solve the initial value problem with a system of differential equations and to describe the behavior of the solution as t approaches infinity. However, the information provided seems to contain some discrepancies, and it is not clear which initial value problem needs to be solved. The initial conditions provided do not match the standard format of an initial value problem. To solve such a problem, typically one would use methods involving eigenvalues and eigenvectors if the system is linear, or numerical methods for non-linear systems. The reference to velocity and acceleration hints that this might relate to a physics context rather than pure mathematics. A more than100words answer would likely involve outlining the method to solve linear differential equations, discussing the eigenvalues, and the long-term behavior of solutions.
Due to the lack of clarity in the problem statement and the potential presence of typos, it is not feasible to provide an accurate solution to the initial value problem as described. If the problem can be clarified or corrected, a more precise solution can be attempted. It is crucial when solving these problems to have the exact system of differential equations and the correct initial conditions.