Final answer:
The problems presented pertain to Differential Equations, requiring the student to calculate general and particular solutions, as well as to plot phase portraits which is essential in the field of mathematics at the college level.
Step-by-step explanation:
The mathematical concept being addressed in the given student question is Differential Equations. The task involves finding the general solution to differential equations, determining a particular solution based on initial conditions, and plotting the phase portrait, all of which are integral to understanding and solving differential equations. This area of mathematics is utilized to express the relationship between a function and its derivatives, often used in a variety of applications including physics, engineering, biology, and economics.
In solving these types of problems, one typically begins by identifying what needs to be determined (the unknowns) and then making a list of what is given or can be inferred (the knowns). After this, the appropriate equation or set of equations is solved for the specified quantity, followed by substituting the known values to obtain a numerical solution. The final solution is checked to ensure it is reasonable and makes sense within the context of the problem.
Phase portraits are graphical representations that show all possible trajectories of a dynamical system in a phase space. They are particularly useful in studying the stability of equilibrium points and understanding the behavior of systems modelled by differential equations.