Final answer:
The value of k for which x(t) = k is a solution to the differential equation depends on the specific context of the equation, which might describe a spring system, chemical kinetics, or another process. To determine k, one must apply given conditions to the relevant equation and solve accordingly.
Step-by-step explanation:
To find the value of k for which the constant function x(t) = k is a solution of the differential equation, we must consider the equation in context. Given various scenarios, k could be a spring constant, a rate constant, or another variable coefficient in a differential equation. For a spring system, k relates the potential energy and position x. For chemical reactions, k could be derived from concentration and rate data. The precise value of k needs to be determined from the given conditions and mathematical expressions presented in the equation. For instance, if x(t) = k is a solution to a harmonic oscillator, then k represents the spring constant, and the function suggests the system is at equilibrium with no movement (since x(t) doesn't change with time).
Without the specificity of the actual differential equation, we can only infer the process of finding k: determine the relation posed by the differential equation, input any initial or boundary conditions, and solve for k.