Coefficients
Homework 1 Due Monday, September 17th at the beginning of class. Show your work. 1. Match the differential equation in (a)-(c) to a family of solutions in (d)-(f). The point of this exercise is not to solve the differential equations in a) - c). (a) y = y 2 (b) y = 1 + y 2 (c) yy = 3x (d) y = tan(x + C) (e) 3x2 − y 2 = C (f) y = −1/(x + C)
2. Find the value of k so that y = e3t + ke2t is a solution of y − 2y − 3y = 3e2t . 3. Solve the following differential equations and IVP’s. You may solve these equations implicitly. (a) y + 3x2 y = x2 (b) y ln t = y+1 t
2
dy dt
2 sin t dy = 0 y (d) y y − t = 0, y(1) = 2, y (1) = 1 (c) cos t dt + 1 + (e) y − y = et y 2 (f) cos(xy) − xy sin(xy) + 2xyex + (ex − 2y − x2 sin(xy))y = 0, y(1) = 0 dy x + 3y (g) = dx 3x + y 2 x sin x dy + (y cos3 x − 1) dx = 0, 0 < x < π (h) cos (i) (x + yey/x )dx − xey/x dy = 0, y(1) = 0 [Hint: Think homogeneous.] 4. Suppose the differential equation dP = (k cos t)P, dt is a model of the human population P (t) of a certain community, where k is a positive constant. Discuss a (non-morbid) interpretation for the solution of this equation. In other words, what kind of population do you think it describes? [Actually solving the equation is not helpful.] 5. Write down a differential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. (Remarks: Consider the forces acting on a falling object, and what they must add up to by Newton’s Second Law. You do not need to solve the differential equation.)
2 2
6. The flu makes its first appearance in Pleasant Island (population 999) when Mrs. Smith’s niece arrives for a visit not feeling well. Let y(t) be the number of infected people. Suppose that apart from the initial visitor, no one enters or leaves the island for an indefinite period of time. Suppose also that the number of islanders infected grows at a rate proportionate both to the number of people infected and the number
2. Find the value of k so that y = e3t + ke2t is a solution of y − 2y − 3y = 3e2t . 3. Solve the following differential equations and IVP’s. You may solve these equations implicitly. (a) y + 3x2 y = x2 (b) y ln t = y+1 t
2
dy dt
2 sin t dy = 0 y (d) y y − t = 0, y(1) = 2, y (1) = 1 (c) cos t dt + 1 + (e) y − y = et y 2 (f) cos(xy) − xy sin(xy) + 2xyex + (ex − 2y − x2 sin(xy))y = 0, y(1) = 0 dy x + 3y (g) = dx 3x + y 2 x sin x dy + (y cos3 x − 1) dx = 0, 0 < x < π (h) cos (i) (x + yey/x )dx − xey/x dy = 0, y(1) = 0 [Hint: Think homogeneous.] 4. Suppose the differential equation dP = (k cos t)P, dt is a model of the human population P (t) of a certain community, where k is a positive constant. Discuss a (non-morbid) interpretation for the solution of this equation. In other words, what kind of population do you think it describes? [Actually solving the equation is not helpful.] 5. Write down a differential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. (Remarks: Consider the forces acting on a falling object, and what they must add up to by Newton’s Second Law. You do not need to solve the differential equation.)
2 2
6. The flu makes its first appearance in Pleasant Island (population 999) when Mrs. Smith’s niece arrives for a visit not feeling well. Let y(t) be the number of infected people. Suppose that apart from the initial visitor, no one enters or leaves the island for an indefinite period of time. Suppose also that the number of islanders infected grows at a rate proportionate both to the number of people infected and the number