Final answer:
It is true that a phase portrait for a dynamical system can be created without finding an explicit solution to the system, by analyzing the stability and behavior around equilibrium points.
Step-by-step explanation:
The statement that a phase portrait can be created without finding a solution for the system is true. A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each point in this plane represents a state of the system, and the phase portrait demonstrates how the system evolves over time from different initial conditions.
Creating a phase portrait does not require the exact solutions of the differential equations; it can be constructed by analyzing the stability of the equilibrium points and the qualitative behavior of the system near these points. This involves using tools such as determining the eigenvalues and eigenvectors of the system's Jacobian matrix at equilibrium points, which can guide us on how solutions behave without actually solving the system.
When setting up equilibrium conditions, various inertial frames of reference and pivot points can be utilized. Some choices can simplify the process significantly. The focus is on understanding the system's behavior rather than finding explicit solutions, which can often be complicated or even unnecessary for this kind of analysis.