Math 217
Exam 1
Page 1
Name:
ID:
Section:
This exam has 16 questions:
• 14 multiple choice questions worth 5 points each.
• 2 hand graded questions worth 15 points each.
Important:
• No graphing calculators! Any non-graphing scientific calculator is fine.
• For the multiple choice questions, mark your answer on the answer card.
• Show all your work for the written problems. You will be graded on the ease of reading your
solution as well as for your work.
• You are allowed both sides of 3 × 5 note “cheat card” for the exam.
1. Linear dependence/independence
Which of the following is a true statement?
I.
Two functions defined on an open interval I are said to be linearly
I provided that one is a constant multiple of the other.
II. Two functions defined on an open interval I are said to be linearly
I provided that one is a constant multiple of the other.
III. Two functions defined on an open interval I are said to be linearly
I provided that neither is a constant multiple of the other.
IV. Two functions defined on an open interval I are said to be linearly
I provided that neither is a constant multiple of the other.
(a) I only.
(b) II only.
(c) III only.
(d) I and II only.
(e) II and III only.
(f) None of the above or some other combination of I, II, III and IV
Solution. Ans. (e)
independent on
dependent on
independent on
dependent on
Math 217
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2. Linear dependence/independence
Consider the following pairs of functions:
I.
II.
III.
IV.
sin x and cos x
ex and xex
3
x and πe x
x and 3x
Which of these pairs of functions is linearly independent on the entire real line?
(a) I only.
(b) II only.
(c) III only.
(d) I and II only.
(e) II and III only.
(f) I, II, and III only.
(g) I, II, III, and IV.
(h) None of the above or some other combination of I, II, III and IV
Solution. Ans. (d)
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3. Linear dependence/independence
Which of the following is a true statement?
(a) The n functions f1 , f2 , . . . , fn are said to be linearly dependent on the interval I
provided there exist constants c1 , c2 , . . . , cn , not all zero, such that
c1 f 1 + c2 f 2 + · · · cn f n = 0
for all x in I.
(b) The n functions f1 , f2 , . . . , fn are said to be linearly independent on the interval I
provided there exist constants c1 , c2 , . . . , cn , not all zero, such that
c1 f 1 + c2 f 2 + · · · cn f n = 0
for all x in I.
(c) The n functions f1 , f2 , . . . , fn are said to be linearly dependent on the interval I
provided there exist constants c1 , c2 , . . . , cn , not all zero, such that
c1 f 1 + c2 f 2 + · · · cn f n = 1
for all x in I.
(d) The n functions f1 , f2 , . . . , fn are said to be linearly independent on the interval I
provided there exist constants c1 , c2 , . . . , cn , not all zero, such that
c1 f 1 + c2 f 2 + · · · cn f n = 1
for all x in I.
Solution. Ans. (a)
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4. Linear dependence/independence
Consider the three solutions y1 (x) = x, y2 (x) = x ln x, and y3 (x) = x2 of the third-order
equation
x3 y (3) − x2 y 00 + 2xy 0 − 2y = 0.
Which of the following statements is true?
(a) y1 , y2 and y3 are linearly independent on the real line.
(b) y1 , y2 and y3 are linearly independent on the interval x > 0.
(c) y1 , y2 and y3 are linearly independent on the interval x ≥ 0.
(d) y1 , y2 and y3 are linearly independent on the interval x < 0.
(e) y1 , y2 and y3 are linearly independent on the interval x ≤ 0.
Solution. Ans. (b)
Math 217
Exam 1
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5. Wronskian
What is the value of the Wronskian of f (x) = x2 , g(x) = x and h(x) = 1?
(a) -3
(b) -2
(c) -1
(d) 0
(e) 1
(f) 2
(g) 3
(h) None of the above
Solution. Ans. (b)
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6. Methods of Solving Differential Equations
What is the value of the Wronskian of f1 (x) = x4 , f2 (x) = x3 and f3 (x) = x2 , f4 (x) = x and
f5 (x) = 1?
(a) 2
(b) 12
(c) 24
(d) 144
(e) 288
(f) None of the above
Solution. Ans. (e)
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7. Complementary Solution
What is the complementary solution of y 00 + y = 3x?
(a) yc = c1 + c2 e−x
(b) yp = 23 x2 − 3x
(c) y = c1 + c2 e−x + 23 x2 − 3x
(d) yc = c1 cos x + c2 sin x
(e) yp = 3x
(f) y = c1 cos x + c2 sin x + 3x
(g) None of the above
Solution. Ans. (d)
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8. Initial Value Problem
Find a particular solution satisfying the initial value problem y (3) − 3y 00 + 4y 0 − 2y = 0;
y(0) = 1, y 0 (0) = 0, and y 00 (0) = 0
(a) 2ex − ex cos x + ex sin x
(b) 2ex + ex cos x − ex sin x
(c) 2ex − ex cos x − sin x
(d) ex (2 − cos x − sin x)
(e) None of the above
Solution. Ans. (d). This is problem #18 from §2.2.
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9. Characteristic Equation
The roots of the characteristic equation of a certain differential equation are 2, -4, 0 (with
multiplicity 2) and 2 ± 4i (with multiplicity 2). Find the DE.
(a) c1 + c2 x + c3 e−4x + e4x (c4 + c5 cos(2x) + c6 sin(2x))
(b) c1 + c2 x + c3 e−4x + e2x (c4 + c5 cos(4x) + c6 sin(4x))
(c) c1 + c2 x + c3 e−4x + e2x (c4 + c5 cos(4x) + c6 sin(4x) + x(c7 cos(4x) + c8 sin(4x))
(d) c1 + c2 x + c3 e−4x + e4x (c4 + c5 cos(2x) + c6 sin(2x) + x(c7 cos(2x) + c8 sin(2x))
(e) c1 + c2 x + c3 e−4x + c4 + c5 cos(4x) + c6 sin(4x) + x(c7 cos(4x) + c8 sin(4x)
(f) c1 + c2 x + c3 e−4x + c4 + c5 cos(2x) + c6 sin(4x) + x(c7 cos(2x) + c8 sin(4x)
(g) None of the above
Solution. Ans. (c)
Math 217
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10. Homogeneous DEs with Constant Coefficients
Find the general solution of the DE
6y (4) + 11y 00 + 4y = 0.
+ c4 sin 4x
(a) y = c1 cos x2 + c2 sin x2 + c3 cos 4x
3
3
(b) y = c1 cos x2 + c2 sin x2 + c3 cos 2x
+ c4 sin 2x
3
3
(c) y = c1 cos √x2 + c2 sin √x2 + c2 cos √2x3 + c4 sin √2x3
4
1
(d) y = c1 e− 3 x + c2 e− 2 x
4
4
1
1
(e) y = c1 e− 3 x + c2 xe− 3 x + c3 e− 2 x + c4 xe− 2 x
(f) None of the above
Solution. Ans. (c). Book problem #17 in §2.3.
Math 217
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11. Undetermined Coefficients
Find the general solution for y 00 + 16y = e3x
(a) y = c1 cos(4x) + c2 sin(4x) +
(b) y = c1 e4x + c2 e−4x +
1 3x
e
25
1 3x
e
25
(c) y = c1 cos(4x) + c2 sin(4x) + 15 e3x
(d) y = c1 cos(4x) + c2 sin(4x) +
1 4x
e
25
(e) y = c1 e4x + c2 e−4x + c3 e3x
(f) y =
1 4x
e
25
+ c2 e−4x +
1 3x
e
25
(g) None of the above
Solution. Ans. (a). Book problem #1 in §2.5.
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Math 217
Exam 1
12. Undetermined Coefficients
Consider the DE y 00 − 2y 0 + 2y = ex sin x. The most appropriate form of the particular solution
yp is
(a) A sin x + B cos x
(b) ex (A sin x + B cos x)
(c) ex x(A sin x + B cos x)
(d) ex x2 (A sin x + B cos x)
(e) ex (x2 + x + 1)(A sin x + B cos x)+
(f) None of the above
Solution. Ans. (c). Book problem #21 in §2.5.
Math 217
Exam 1
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13. Initial Value Problem
Solve the initial value problem y 00 + 3y 0 + 2y = ex ; y(0) = 0, y 0 (0) = 3
(a)
(b)
(c)
(d)
(e)
(f)
(g)
15 −x
e − 83 e−2x − 61
6
15 −x
e − 43 e−2x − 61 ex
6
15 −x
e − 83 e2x − 61 ex
6
15 x
e − 68 e−2x − 16 ex
6
15 −x
e − 86 e−2x − 61 ex
6
15 −x
e + 83 e−2x − 61
6
15 −x
e − 43 e−2x + 61 ex
6
(h) None of the above
Solution. Ans. (h). None of the above. The solution is 16 (15e−x − 16e−2x + ex )
Math 217
Exam 1
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14. Variation of Parameters
In the derivation of the formula for the method of variation of parameters for second-order
linear nonhomogeneous equations, we encountered the system of equations
u01 y1 + u02 y2 = 0
u01 y10 + u02 y20 = f (x).
Let W (x) be the Wronskian of y1 and y2 . What are the solutions of u1 and u2 ?
(a) u1 =
R
y1 (x)f (x)
W (x)
dx and u2 = −
R
y2 (x)f (x)
W (x)
dx .
(x)f (x)
(x)f (x)
(b) u1 = − y2W
dx and u2 = y1W
dx.
(x)
(x)
R y2 (x)f (x)
R y1 (x)f (x)
(c) u1 =
dx and u2 =
dx.
W (x)
W (x)
R y2 (x)f (x)
R y1 (x)f (x)
(d) u1 =
dx and u2 = −
dx.
W (x)
W (x)
R y1 (x)f (x)
R y2 (x)f (x)
dx and u2 = −
dx.
(e) u1 = −
W (x)
W (x)
R
R
(f) None of the above
Solution. Ans. (b). These are the coefficients of the formula for variation of parameters.
WRITTEN PROBLEM—SHOW YOUR WORK
Math 217
Exam 1
Name:
ID:
Section:
Note: You will be graded on the readability of your work. Use the back of this sheet, if
necessary.
15. Methods of Undetermined Coefficients
Use the method of undetermined coefficients to find the general solution of
y 00 − y 0 − 2y = 6x + 6e−x
WRITTEN PROBLEM—SHOW YOUR WORK
Math 217
Exam 1
Name:
ID:
Section:
Note: You will be graded on the readability of your work. Use the back of this sheet, if
necessary.
16. Applications One solution of the DE
6y (4) + 5y (3) + 25y 00 + 20y 0 + 4y = 0
is y = cos(2x). Find the general solution.