Consider the dynamical system represented by the following differential equation 3 dx dt + 2x = 1. (a) Write the characteristic equation for the system (b) Find time constant τ, the system eigenvalue λ, and the time to half amplitude Th (if the system is stable), or to double amplitude Td (if it is unstable). (c) Find the complementary (homogeneous) solution of the differential equation, x(t). (d) What is the particular solution? (e) Given that x(0) = 1, what is the total solution of the above differential equation.
Suppose a company's profit (in dollars) is given by P(x) = 230x − 0.3x2 − 5,200, where x is the number of units. Find P′(300). Interpret P′(300). The marginal profit is $ per unit. The profit on the 301st unit is $ . Find P′′(300). Interpret P′′(300). The marginal profit decreases at a constant rate of P′′(300) per unit per unit. The marginal profit increases at an increasing rate of P′′(300) per unit per unit. The marginal profit increases at a decreasing rate of P′′(300) per unit per unit. The marginal profit increases at a constant rate of P′′(300) per unit per unit. The marginal profit decreases at a decreasing rate of P′′(300) per unit per unit.