Math Questions
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MATH 2255, Fall 2020
Homework 2
Due Friday, September 11, 11:30am
.
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(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 < ∞.