Math 224 (Differential Equations and Linear Algebra)

As a sample and as a means for you to prepare for this exam, the following are some sample problems from prior

semesters. This sample is reflective of the material we covered, but is NOT INCLUSIVE. It is STRONGLY advised that

you study your lecture notes and the homework problems.

Sample Problems:

1. Determine the order of

22

2

x yd y dye x

dx dx+ = and state whether the equation is linear or not

2. Solve 2 2

8cos csc 0y dx x dy+ = subject to 12 4

( )yπ π

=

3. Find the values of “r” for which ( )r x

y x e= solves the equation 3 4 0y y y′′ ′+ − =

4. Solve (2 ) 1ye x y′− = by considering x as a function of y [Hint: Find x(y)] 5. Solve

2

2 ( 1)

3

dy x y

dx x

−=

+

6. Solve2

5(4 )

dxx

dt t+ =

−, subject to (0) 4x = 7. Orthogonal trajectories problems. (Refer to your notes)

8. Directional field problems. (Problem 5, Sec.2.1) 9. Find all z satisfying the equation 4

64 0z + = (z = Complex No.)

10. The fundamental set of solutions of a given differential equation. (Problem 39, Sec.3.3) 11. Existence & uniqueness problems. (Problems 3, 9; Sec.3.1 and problems 3, 13: Sec. 2.2) 12. Find all values of (x, y) where the theorem of existence and uniqueness theorem guarantees a unique solution for

3(tan )y x y x y x′′′ ′+ + =

13. Linear dependency, independency, and the Wronskian for set of functions.(Sec. 3.2)

14. Given 1y x= is one of the fundamental solutions of 5 0y y′′′ ′′+ = , find the other solution.

15. Linear differential operators (Problems 11, 17, 21, and 31on Sec. 3.1)

16. At 2 p.m. (t = 0) on a cool (034 F) afternoon in March, Sherlock Holmes measured the temperature of a dead body

to be 038 F. One hour later the temperature was

036 F. After a quick calculation using Newton’s law of cooling, and taking

the normal temperature of a living body to be 098 F, Holmes concluded that the time of death was 10 a.m. Was Holmes right?

17. A tank initially contains 200 L of solution in which is dissolved 100 g of chemical. A solution containing 0.5 g/L of the chemical flows

into the tank at a rate of 6 L/min, and the well-stirred mixture flows out at a rate of 4 L/min. determine the amount of chemical in the tank after 60 minutes.

18. Consider the RL circuit which has R = 2 ohms, L =2

3 H, and E (t) = 10 sin 4t volts. If there is no current flowing initially, determine

the current i(t) in the circuit for 0t ≥ .

19. Some “True/False” questions.