Exam 8: First-Order Differential Equations

Is the following differential equation separable or not?

Suppose the income tax structure is as follows: the first $28,000 is taxed at 20%, the remainder is taxed at 30%. Compute the tax on an income of $40,000. Now suppose that inflation is 5% and you receive a cost of living (5%) raise to $42,000. Compute the tax on this income. To compare the taxes you should adjust the tax for inflation (add 5%).

Write the following second-order equation as a system of first-order equations.

Find the solution to the following separable differential equation.

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

Find all equilibrium points.

The University of XYZ has a goal to increase its endowment from the initial value of $100,000,000, to $150,000,000 over 4 years. If the interest rate earned by the endowment (after expenses) is 5% each year (compounded continuously), and the contributions become available continuously and at a constant rate, how much will they actually have to collect from contributors over those 4 years to meet their goal?

Is the following differential equation separable or not?

Consider a chemical system containing species A, B, and C; and that A and B can react to make C in a bimolecular reaction with rate constant of k₁, and C can decompose to make A and B in a first order reaction with rate constant of k_-1. If the instantaneous amounts of A, B, and C are represented as a, b, and c, and the initial amounts are given as A₀, B₀, and C₀, the change in C can be represented with the differential equation . If A₀ = 2 , B₀ = 2, C₀ = 0, k₁ = 0.04, and k_-1 = 0.04, solve the differential equation for c and graph the solution. [Note: c can never be larger than C₀ plus the smaller of A₀ or B₀. Nor can it be smaller than 0.]

Find all equilibrium points for the following system of equations.

Identify the equilibrium solutions for , and determine if they are stable or unstable.

Identify the equilibrium solutions for , for , and determine if they are stable or unstable.

Is the following differential equation separable or not?

A stirred tank with volume, V, has a feed stream of concentrated floor cleaner flowing into it at a rate of f . The flow stream has a concentration of c_i. The outlet stream also flows at f but with a concentration of c, the concentration of the floor cleaner solution in the tank. The rate of change of the concentration is proportional to the difference between c_i and c with proportionality constant of f/V: . If the tank volume is 2000 L, the flow rate is 6 L/min, the inlet concentration is 0.9, and the initial concentration of the floor cleaner in the tank is 0.0, how long until the concentration in the tank is 0.7?

Find the solution of the differential equation, y' = -2y, satisfying the initial condition, y(3) = 0.

Solve the following initial value problem explicitly. y(1) = -3

Use the direction field below to sketch a solution curve and estimate the initial value y(0) for the differential equation , such that the solution curve passes through the point (2, 2).

Match the appropriate slope field with the differential equation .

The rate at which water flows out of a drain in the bottom of a certain tank is proportional to the height of water in the tank. The tank is a vertical cylinder with cross-sectional area of 1.0 m², so that every 1 cm in height represents 10 L. If the flow is 10 L/min (i.e. 1 cm/min) when the water level is 700 cm, how long will it take for the level to go from 700 cm to 30 cm?

The text describes logistic growth with an equation for the actual population in terms of a growth constant and a maximum population (carrying capacity), . The equation could also be written for a fraction of the maximum population in terms of a fractional growth constant. Re-express the differential equation in terms of the fractional population, . Compare the time it takes for the population to go from 60% of the maximum to 80% of the maximum with the time it takes to go from 80% of the maximum to 90% of the maximum, if k = 0.00100 day^-1 and M = 4000?