MATH 241 (ORDINARY DIFFERENTIAL EQUATIONS) TEST 1 PRACTICE
PROBLEMS (COHEN)
General Comments and Advice: The student should regard this review sheet only as a sample of
potential test problems, and not an end-all-be-all guide to its content. Anything and everything which
we have discussed in class is fair game for the test. The test will cover Sections 1.1, 2.1, 2.2, 2.3, 2.5,
2.7, and 3.1. Don’t limit your studying to this sheet; if you feel you don’t fully understand a particular
topic or technique, then do more problems out of your textbook!
Major Facts About the Test:
(1) There will be four problems, although some may have multiple parts. This gives you 15 minutes
per problem. The problems are worth 15 points each for a total of a 60 point exam.
(2) Calculators, cell phones, computers, tablets, notes, and books are not allowed on the exam. I
will provide a handout with descriptions of important techniques for your use during the exam.
The handout is included at the end of this document.
1. Find all solutions to the following first-order differential equations.
(a) y0= 8x7y
(b) y0+xcos2y= 0
(c) y0=1+2x
y2+y2x2
(d) y0=x2y−y+x2−1
(e) y0+6t5
t6+ 1y=t
(f) 3y0−y= (3e−tsin t)y4
(g) t3+y3−ty2y0= 0, t > 0
2. For each differential equation in the previous problem, find a solution which satisfies the initial
condition given below (in respective order).
(a) y(1) = 10
(b) y(0) = π
4
(c) y(0) = 3
(d) y(0) = 0
(e) y(2) = 1
(f) y(π
2)=1
(g) y(e) = e
Also look at: 2.1 #1–12; 2.2 #1–16, 25–31; 2.4 #23–25
3. Suppose a tank holds V0gallons of a brine solution. Another brine solution, with concentration S1
pounds of salt per gallon, is allowed to flow into the tank at a rate R1gallons per minute. A drain at
the bottom of the tank allows the mixture in the tank to flow out at a rate of R2gallons per minute.
We assume the brine flowing out is well-mixed. If yrepresents the number of pounds of salt in the tank
after tminutes, then yis modeled by the ODE:
y0=(Rate salt enters the tank) −(Rate salt leaves the tank) = S1R1−y(t)
V(t)·R2,
1
