Final answer:
To find the equation of the orbits, we'll first solve each system of differential equations and then integrate the resulting equations. Let's go through each system: To answer the student's question, we divide one differential equation by the other and integrate to find the orbit equations. Then, for general solutions, we solve the systems using methods like eigenvectors and eigenvalues for component and vector forms.
Step-by-step explanation:
To solve the given system of differential equations, we can divide one equation by the other and integrate to find the relationship between x and y that describes the orbits in the phase plane.
Secondly, to find the general solutions in component and vector forms, we will solve the systems of differential equations directly.
Each system represents a different type of dynamical behavior and requires a tailored approach to finding the solutions.
a) System: x' = -3y, y' = 2x
By dividing the two equations, dy/dx = -(2/3)x/y, we can integrate to find the orbit equation. The component form and vector form solutions can be found using methods such as eigenvectors and eigenvalues.
b) System: x' = -2y, y' = -4x
Dividing and integrating this system will yield another orbit equation. The solutions in this case might involve trigonometric functions due to the rotational nature of the system.
c) System: x' = -3x, y' = 2y
This system describes exponential growth and decay. The general solution will involve exponential functions of time for both x and y components.
d) System: x' = 4y, y' = 2y
The y' equation suggests exponential growth for the y component, while the x equation involves the y component. Solving this system will also involve exponential functions.