Consider an owl population in which P represents hundreds of owls and t is given in years. Say we know from previous data collection efforts that:The only equilibrium solutions that exist are at P(0), P(4), and P(8). If the value of P is 9 , the amount of owls decreases. If the value of P is 3, the amount of owls increases. (a) Assuming we are not concerned with what happens when we start with "negative" owls, sketch two different possible phase lines that could represent this population, labelling your equilibrium solutions as stable, unstable, or semistable. (b) Describe in this context what it means for an equilibrium solution to be unstable. (c) Say instead the population was modelled by dPdt = 0.2(P−3)(P−2). Determine the equilibrium solutions for this equation, and state whether they are stable, unstable, or semi-stable.
Solve the differential equations below by choosing an appropriate method. (a) t2dy dt = 3t3 − 4y (b) d2y dt2 − 14dy dt + 49y = 0 (c) d2y dt2 + 64y = 0 (d) d3y dt3 − 3d2y dt − 45dy dt − 81y = 0
Recall that a logistic growth model takes the form dP dt = rP(1 − P K) where r is some growth or decay constant, P is the number in the population, and K is the carrying capacity. Suppose data has been collected on the population counts of different types of fish in Horsetooth Lake, and is shown below: Scientists suspect that these populations follow a logistic growth model. They know that Horsetooth's carrying capacity for Walleye is 500 fish, the carrying capacity for Bluegill is 600 fish, and the carrying capacity for small-mouth bass is 1000 fish. (a) Determine the growth/decay constant (r) for one of the species of fish using the data provided. Specify which species you have chosen. (hint: recall the form that the solution to a logistic growth model takes, and use log rules to isolate r!)
For each of the following, find a differential equation with the given solution. (Hint: Use what you know about characteristic polynomials) (a) y(t) = c1e−t + c2e−4t (b) P(t) = c1e2t + c2te2t (c) y(t) = c1cos(5t) + c2sin(5t) (d) P(t) = c1et + c2e−2t + c3e6t
Write a differential equation that meets the given criteria: (a) A second-order differential equation that we could use a characteristic polynomial to solve. (b) A second-order differential equation that we would need the method of undetermined coefficients in order to solve. (c) A second-order differential equation for which neither a characteristic polynomial nor the method of undetermined coefficients would yield a solution. (d) Would the characteristic polynomial method work if applied to a first-degree homogeneous differential equation with constant coefficients? Explain why or why not.
Determine an appropriate Method of Undetermined Coefficients "guess" for the particular solution, yp, given the following second-order differential equations. You do not need to solve. (a) y′′ + 3y′ + 5y = 3e−3t (b) 2y′′ − 3y′ + 6y = cos(πt) + t2 (c) y′′ + 3y′ + 5y = 5t3e6t (d) 2y′′ − 3y′ + 6y = 3e4tsin(5t) + 5
Find the generalized solution (yh + yp) for the following differential equation using the Method of Undetermined Coefficients. (Hint: recall that yp must be linearly independent [be a fundamentally different term] from the terms found in our yh, the homogenous solution) y′′ − 6y′ + 9y = 9e3t
The charge present in a circuit capacitor at time t is given by Q(t). The rate of change of the charge in the capacitor, dQ dt, then represents the current running through the circuit. By applying Faraday's Law, Ohm's Law, and Kirchoff's Law, we can describe the behavior of the circuit using the second-order differential equation LQ′′ + RQ′ + 1/CQ = E(t) where L is an inductance constant (in units of henrys), R is the resistance (in units of ohms), and C is the constant of capacitance for the capacitor (in units of farads). E(t) is some function in terms of t representing the voltage of the battery. Consider the case in which we have a circuit with an inductance constant of 2 henrys, a resistance of 6 ohms, and 0.25 is our constant of capacitance. The voltage of the battery can be represented by the function 8 sin(2t). (a) Write out a differential equation (using the general form given above) to represent this circuit system. (b) Solve this differential equation as though it were homogeneous (find Qh for this instance). (c) Use the Method of Undetermined Coefficients to identify the particular solution Qp for this instance. (d) What is the generalized solution to your differential equation given in part (a)? (e) Briefly explain (1-2 sentences) what the generalized solution represents in this problem's context.