Exam 2 Review Guide, Blank

M A T H 083, E X A M 2 R E V I E W G UI D E Sections 3.1-3.6, 4.1-4.4 1. Statements a – e below refer to the second order non-homogeneous L [y] = g(t) with particular solution p(t). The fundamental set of the corresponding L [y] = 0 is { y1 (t ), y 2 (t )} on some interval I. a. T or F

y(t) = 0 is always a solution of L [y] = 0.

b. T or F The Wronskian of {5 y1 (t ), 3 y 2 (t )} will never equal zero on I. c. T or F

L [y1(t)] = g(t).

d. T or F

L (c1 y1 (t ) c 2 y 2 (t ))

e. T or F

y1(t) and y2(t) are linearly-independent on I.

g (t ) .

2. Check ( ) the differential equations below for which you could use undetermined coefficients to find Y(t). a. [ D 2 2 D 3] y t 1 ___________ b. [t 2 D 2 2t D 3] y 4 ___________ c. [ D 2 4] y tet sin3t ____________ 3. On what intervals will the solutions of [(t 1) D 2

ax by 0 a b and det cx dy 0 c d system?

4. a) If

ax by e , e and/or f cx dy f solution of the system? b) If

D] y

t 4 have their domains?

0, what do we know about the solution of the

0, and det

a b = 0, what do we know about the c d

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5. Consider L t 2 D 2 2tD 4 a. Find L [y] if y (t ) t 3 .

b. Based on your answer to part a, y (t )

t 3 is a solution of L [y] = _________.

1

6. Find all roots of ( 1) 3 .

7. The fundamental set of the fourth order L [y] = 0 is {1, t, cos 2t, sin 2t }. a. Evaluate the Wronskian of {1, t, cos 2t, sin 2t }.

b. What is/are the interval(s) on which {1, t, cos 2t, sin 2t } form a linearlyindependent set of solutions of L [y] = 0? ____________________ 8. Short answer. Fill in the blanks. a. By definition, the functions f1 (t ), f 2 (t ), f 3 (t ) are linearly independent on some interval I if the relationship __________________________is true if and only if _____________________. b. The functions f1(t) = e 2 t and f2(t) = e 3t are linearly _________________ on ( , ). c. The functions f1(t) = e 2 t and f2(t) = 3e2t are linearly _________________ on ( , ).

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8. For r = 1 2i use the Cauchy-Euler identity to express e r in terms sin and cos.

9. If one solution of t 2 y' ' 5ty' 4 y find y2(t).

0 (t > 0) is y1 (t )

t 2 , use reduction of order to

10. Find the general solution of [ D 3 ( D 1) 3 ( D 2 2 D 8) 2 ] y 0 .

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11. ( D 2

4) y

4e 2 t

3t has a particular solution with the form

Y (t ) k1 k 2 t k 3 e 2t . a. Find the values of the coefficients k1, k2, and k3.

b. What is the general solution of ( D 2

4) y

4e 2 t

3t ?

12. Consider [ D 2 ( D 3)] y = 5te3t 3et cos 2t . Find the form of Y(t) using the method of Undetermined Coefficients. DO NOT SOLVE FOR THE COEFFICIENTS.

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13. Consider [t 2 D 2 t D 1] y 3t 2 on (0, ) for which the fundamental set of L[y] = 0 is {t, t-1}. Use the method of Variation of Parameters to find a particular solution, Y(t), of [t 2 D 2 t D 1] y 3t 2 . a. What is the form of Y(t)? b. What system of equations must be solved to completely find Y(t)?

c. Complete the work necessary to find Y(t).

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