1. Find and classify (as stable, unstable, or semistable) all equilibrium solutions for the following differential equations. Summarize your work in a phase line. (a) y' = y(3 - y)(25 - y?) (b) y' = Iny - 2, y >0 (c) y' = 5 y + 2' y 20 2. The Fitzhugh-Nagumo model for the electric impulse in a neuron states that, in the absence of relaxation effects, the electric potential, v(t), in a neuron obeys the differential equation at = - vv2 - (1 + a)u+ a) where a is a constant such that 0 0 represent some population. Suppose the rate of change of the population is proportional to the population. This statement is expressed mathematically by the equation: dP dt = kP (a) Find equilibrium points and construct phase lines to determine values of the non-zero constant k for which i. P(t) is increasing ii. P(t) is decreasing MATH 141 Homework 7 (b) Use separation of variables to solve the differential equation in terms of k to verify your above responses. 4. A modification of the above differential equation is dy = k(y - a) dt where k # 0 and a are constants. (a) Which values of k is the equilibrium point stable? Evaluate lim y(t). (b) Which values of k is the equilibrium point unstable? Evaluate lim y(t). Hint: there is no need to solve for y(t) to evaluate the above limits. 5. A model for the velocity of a falling object with mass m is the differential equation du dt k (v+ mg) Acceleration due to gravity is g > 0 and k # 0 is the air resistance coefficient. (a) Find and classify equilibrium points using a phase line. (b) Sketch solution curves based on your phase line. (c) Use separation of variables and find the solution satisfying the initial condition v(0) = 0. (d) Evaluate and interpret lim v(t). (e) How long will it take for the velocity of the falling object to reach -mg/2k? 6. A cup of coffee has a temperature of 80C when it is placed outside where the temperature is 20 C. Suppose the temperature of the coffee, T(t), changes according to Newton's Law of Cooling with proportionality constant k = 0.09. (a) Find and classify the equilibrium solution(s) using a phase line. (b) Find T(t). (c) Evaluate and interpret lim T(t). (d) How long will it take for the coffee to reach a temperature of 25 C? How long to reach a temperature of 20 C? 7. A corporation maintains a balance of B(t) (in millions of dollars) in an account, where t is measured in years. Interest with a nominal annual rate of 2% compounds continuously (that is, the rate at which interest is added to the account is proportional to B with proportionality constant 0.02 year"). In addition, the corporation continuously withdraws money from the account at a rate of 4 million dollars per year. (a) Write a differential equation for the balance B(t). (b) Find all solutions to your differential equation. (c) Suppose that B(0) = 300. Determine lim B(t) and interpret your result. oo+ 1 (d) Suppose instead that B(0) = 100. When will the account be empty
