MATH 331.2, Fall 2008: Practice problems
These problems are selected problems from quizzes and midterms in the last few
years.
1. Solve the initial value problem dy
dt =−3y−cos(2t), y(0) = 4.
2. Solve the initial value problem dy
dt =−et/y , y(0) = −2.
3. Consider the differential equation dy
dt =y2−4y+ 2 .
(a) Find all the equilibrium points, determine their type (sink, source, node).
(b) Draw the phase line of the system.
(c) Sketch the solutions of the equation with the initial conditions (a) y(0)= 1, (b)
y(0)=-5, and (c) y(0)=6. Indicate clearly the behavior for t→ ∞ and t→ −∞.
4. A home buyer can afford to spend no more than $1000 per month on mortgage
payments. Suppose that the interest rate is 5% (per year) and that the term of the
mortgage is 20 years. Assume that interest is compounded continuously and that
payments are also made continuously.
(a) Determine the maximum amount that this buyer can afford to borrow.
(b) Determine the total interest paid during the term of the mortgage
5. Solve the initial value problem dx
dt =x2[t+ sin(t)] with x(0) = 6.
6. Mary initially deposits $1000 in a savings account that pays interest at the rate
of 5% per year (compounded continuously). She also arranges for $25 per week to
be deposited automatically into her account.
(a) Assume that weekly deposits can be approximated by continuous deposits. Write
down an initial value problem for her account balance S(t) over time (tmeasured
in years).
(b) How long does she needs to save to buy a $5000 car?
7. Consider the following equation for a certain population of squirrels given by
P(t) (tis measured in years).
dP
dt = 2P1−P
2(P−1)
(a) Find all the equilibrium points of the equations. Draw the phase line and indicate
the type of each equilibrium points (i.e., sink, source, or node).
(b) Make a graph of the solutions with initial conditions P(0) = 1/4, P(0) = 3/2,
and P(0) = 3.
