Final answer:
Systems (A) and (D) are nonlinear autonomous, (B) is linear not autonomous, and (C) is linear autonomous.
Step-by-step explanation:
The question asks to identify each of the given systems as linear or nonlinear, autonomous or not autonomous. A system is considered linear if it can be written in the form of a first-order linear differential equation, which is an equation involving the first derivative of the function and possibly the function itself, but not any higher powers or other operations on these terms. An autonomous system is one in which the derivative of the function depends only on the function itself and not explicitly on the independent variable, which is often represented by 't' for time.
- (A) dx/dt = x2 is nonlinear because the term x2 is a higher power of the function x, and it is autonomous because the derivative of x depends only on x and not directly on t.
- (B) dy/dt = sin(t) + y is linear because it only involves the first derivative of y and the function y itself, without higher powers or other operations. However, it is not autonomous because the derivative of y depends explicitly on t, as indicated by the sin(t) term.
- (C) dz/dt = 3z - 2 is linear because the terms involve only the first derivative of z and the function z itself, and it is autonomous because the derivative of z does not directly depend on t.
- (D) du/dt = u(1/2) is nonlinear because of the square root operation on the function u, and it is autonomous as the derivative depends only on the function u itself.