Math 215
§2.1: 13, 15, 30,
Spring 2010
§2.2: 3, 10, 21, §2.3: 2, 8, 16,
• (§2.1 #13) Solve y 0 − y = 2te2t ,
Assignment 1
§2.4: 1, 15, 22(a,b)
y(0) = 1.
• (§2.1 #15) Solve ty 0 + 2y = t2 − t + 1,
y(1) = 12 ,
t > 0.
• (§2.1 #30) Find the value of y0 for which the solution of the initial value problem
y 0 − y = 1 + 3 sin t,
y(0) = y0
remains finite as t → ∞ (i.e., uniformly bounded by some constant for all t).
• (§2.2 #3) Solve y 0 + y 2 sin x = 0.
• (§2.2 #10) Consider
y 0 = (1 − 2x)/y,
y(1) = −2.
(a) Find the solution in explicit form.
(b) Plot the graph of the solution.
(c) Determine the interval in which the solution is defined.
• (§2.2 #21) Solve
y 0 = (1 + 3x2 )/(3y 2 − 6y),
y(0) = 1
and determine the interval in which the solution is valid.
Hint: look for points where the integral curve has a vertical tangent.
• (§2.3 #2) A tank initially contains 120 L of pure water. A mixture containing a
concentration of γ g/L of salt enters the tank at a rate of 2 L/min, and the wellstirred mixture leaves the tank at the same rate. Find an expression in terms of γ for
the amount of salt in the tank at any time t. Also find the limiting amount of salt in
the tank as t → ∞.
• (§2.3 #8) A young person with no initial capital invests k dollars per year at an annual
rate of return r. Assume that investments are made continuously and that the return
is compounded continuously.
(a) Determine the sum S(t) accumulated at any time t.
(b) If r = 7.5%, determine k so that $1 million will be available for retirement in 40
years.
(c) If k = $2000/year, determine the return rate r that must be obtained to have $1
million available in 40 years.
• (§2.3 #16) Newton’s law of cooling states that the temperature of an object changes
at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton’s law of
cooling. If the coffee has a temperature of 200 ◦ F when freshly poured, and 1 min
later has cooled to 190 ◦ F in a room at 70 ◦ F, determine when the coffee reaches a
temperature of 150 ◦ F.
• (§2.4 #1) Determine (without solving the problem) the interval of validity of the
following problem
(t − 3)y 0 + (ln t)y = 2t, y(1) = 2.
• (§2.4 #15) Solve the problem
y 0 + y 3 = 0,
y(0) = y0 ,
and determine how the interval in which the solution exists depends on the initial
value y0 .
• (§2.4 #22(a,b)) (a) Verify that both y1 (t) = 1 − t and y2 (t) = −t2 /4 are solutions of
the initial value problem
y0 =
−t + (t2 + 4y)1/2
,
2
y(2) = −1.
Where are these solutions valid?
(b) Explain why the existence of two solutions of the given problem does not contradict
the uniqueness part of Theorem 2.4.2.
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