3320 diff equa
HW 4 DIFF EQ 0. Read chapters 3.1 and 4.1 1. A contagious disease is spread throughout a community. Let x(t) be the number of people have contracted the disease and y(t) be the number of who have not yet been exposed. Suppose a small community has a fixed population of n people. If one infected person is introduced into this community, then it can be argued that x(t) and y(t) are related by x + y = n + 1. If we assume that the rate dx/dt is jointly proportional to x and y then we arrive at the differential equation dx = kx(n + 1 − x), x(0) = 1 dt (a) Determine a nontrivial constant solution, call it xc . (b) What happens if the total infected population exceeds xc ? (c) What happens if the total infected population is below xc ? (d) Sketch the slope field of the differential equation. 2. In Cleaning Lake Ontario, suppose that pollution input into the lake suddenly ceases and that the pollution present in the lake at time t is always well mixed but decreases as fresh water flows into the lake and contaminated water flows out of the lake. Assume that fresh water enters the lake at the same rate that contaminated water leaves the lake; denote this flow rate by r (measured in cubic meters per hour) (a) If the rate at which the pollution level decreases equals the difference between the rate at which the pollution enters the lake and leaves the lake, what is the differential equation for x(t)? (b) Let x(t) denote the pollution concentration (number of gallons per cubic meter of lake water) at time t, and let V denote the constant volume of the lake. What is the total pollution level in the lake at time t? (c) At what rate is pollution carried out of the lake? 3. Newton’s law of motion states that F = ma where F is the forced acting on an object, m is the mass of the object, and a is the acceleration. Now suppose a rock is tossed upward from the roof of a building as illustrated in the Figure below. 1 2 The acceleration of the rock is given by the second derivative ddt2s . If we assume that the upward direction is positive and that no force acts on the rock other than the force of gravity, then Newton’s second law gives m d2 s = −mg . dt2 Suppose the object experiences drag which is proportional to the square of the velocity. Let the projectile be propelled vertically upward and let s(t) denote the altitude above the ground. (a) What is the drag force acting on the projectile as it moves? (b) What is the differential equation that models the projectile’s altitude s at time t? 4. Find the solution to each second order linear differential equation by finding the roots of the corresponding characteristic equation. (a) y ′′ − 3y ′ + 2y = 0 (b) y ′′ + 5y ′ + 4y = 0 5. A solution y1 is given for each of the following 2nd order linear differential equations y ′′ + P (x)y ′ + Q(x)y = 0 . Find a second solution using the reduction of order formula Z y2 (x) = y1 e− R P (x)dx (y1 )2 dx (a) xy ′′ + y ′ = 0, y1 = ln(x) (b) (1 − x2 )y ′′ + 2xy ′ = 0, y1 = 1 (c) (1 − 2x − x2 )y ′′ + 2(1 + x)y ′ − 2y = 0, y1 = x + 1 6. Differential Equations of the form ay ′′ + by ′ + cy = 0 with a, b, c constants always possess one solution of the form y1 = erx . Explain why this statement does not contradict the fact that The differential equation y ′′ + 9y = 0 has solution y1 = sin(3x). 2
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