Math Questions

MATH 2255, Fall 2020 Homework 2

Due Friday, September 11, 11:30am.

Please upload your homework on Carmen. Late homework is not accepted. I encourage you to work with others on homework problems, but you must write up your own solutions. Solutions must be presented clearly, or will be marked down.

(1) In the following initial value problems, determine (without solving the problems) an interval in which the solution is certain to exist. (a) (t− 3)y′ + (ln t)y = 2t, y(1) = 2; (b) (4 − t2)y′ + 2ty = 3t2, y(−3) = 1; (c) (4 − t2)y′ + 2ty = 3t2, y(1) = −3; (d) (ln t)y′ + y = cot t, y(2) = 3.

(2) In the following ODEs, state where in the ty-plane the hypotheses of the existence and uniqueness of solutions theorem (for nonlinearn ODEs) are satisfied. Then describe the possible y0 in the initial condition y(0) = y0 in order for the ODE to have a unique solution on the interval (−h, h) for some h > 0. (a) y′ = t−y

2t+5y ;

(b) y′ = (t2 + y2)3/2;

(c) y′ = (cot t)y 1+y

.

(3) Solve for the continuous solution of the initial value problem

y′ + 2y = g(t), y(0) = 0,

where

g(t) =

{ 1, 0 ≤ t ≤ 1 0, t > 1.

(4) Suppose that there are five rabbits in an enclosed pasture, and let P (t) model the change of population with respect to the time t. If the birth rate (number of births/rabbit per unit time) were bP and the death rate is 0 (i.e. the rabbits do not die), then describe what happens to the population of the rabbits. [Hint: (1) In class, the birth rate of a simple population model was b; (2) This scenario is sometimes called the “population explosion.”]

(5) In each problem (with initial condition y(0) = y0), determine the critical points, and classify each one asymptotically stable, unstable, or semistable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. (a) dy/dt = −k(y − 1)2, k > 0,−∞ < y0 < ∞; (b) dy/dt = ay − b

√ y, a > 0, b > 0, y0 ≥ 0;

(c) dy/dt = y2(1 −y)2, −∞ < y0 < ∞.