Name:
MATH 250
Student Number:
Midterm Exam II
Instructor:
Sample 3
Section:
Sample 3
This exam has 9 questions for a total of 100 points. Read all directions.
In order to obtain full credit, all work must be shown.
Check that your exam has all 9 questions.
You may not use a calculator, cell phone, or any notes on this exam.
This exam booklet will be collected at the end of the exam.
Do not write in this box.
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MATH 250 Exam II-Sample3
1. (8 points) For full credit you must write the full word ”True” or ”False” in the space provided.
a)
Z
∞
te−(s−8)t dt
0
The following 2 questions refer to the internal above marked ”a)”
The integral represents the Laplace transform of f (t) = t .
The integral converges for all values of s.
b) x00 + x = 4 cos t
The following 2 questions refer to the ODE above marked ”b)”
The spring-mass system modeled by the ODE will oscillate at frequency ω = 1.
The spring mass system will pass through the equilibrium position
(x = 0) infinitely many times.
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MATH 250 Exam II-Sample3
2. (12 points) Find the form of the particular solution yp to each ODE. You do NOT need to solve
for any constants. You do NOT need to write the general solution.
a) y 00 + 16y = 4te−2t
b) y 00 + 5y 0 + y = 3t2 − 1
c) y 00 + 2y 0 + y = et − te−t
d) y 00 + 4y = 6 cos t
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MATH 250 Exam II-Sample3
3. (12 points) Use the method of undetermined coefficients to find the general solution to the ODE:
a)y 00 − 3y 0 − 4y = 8
b) y 00 + 25y = 17et
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MATH 250 Exam II-Sample3
4. (12 points) A mass of 1 kg is tied to the end of a spring and stretches it 5 m. The damping force
of the spring-mass system is numerically equal to the velocity. The mass is set in motion from the
equilibrium position with an initial downward velocity of 1 m/s . Use the value g = 10 m/s2 for
the gravitational acceleration g.
a) Write an ODE with initial condition to describe the position of the mass at any time t.
b) Solve the ODE with initial condition (Initial Value Problem) from part a)
c) Find the first two times t ≥ 0 when the mass passes through equilibrium.
d) Find the time t when the mass reaches its lowest point.
e) What is the position when it reaches its lowest point?
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MATH 250 Exam II-Sample3
5. (12 points) Consider the ODE y 00 + 25y = 0
a) Determine the interval of convergence for any power series solution about x0 = 0?
b) Find the recursion relation for a power series solution about x0 = 0 to the ODE above, and
clearly state for which index values it is valid.
c) Show by direct substitution that
y=
∞
X
(−1)n (5x)2n
n=0
(2n)!
is a solution to this ODE.
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MATH 250 Exam II-Sample3
6. (12 points) Find the first 3 non-zero terms of each of 2 linearly independent power series solutions
about x0 = 0 to the ODE below. You may assume c0 and c1 are non-zero.
(1 − x)y 00 + y = 0
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MATH 250 Exam II-Sample3
7. (8 points) Use the Laplace transform to solve the initial value problem:
y 00 + 2y 0 + 2y = δ(t − π)
y(0) = 1, y 0 (0) = 0
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MATH 250 Exam II-Sample3
8. (12 points) Use the Laplace transform to solve the initial value problem:
y 00 + 4y = U(t − π) − U(t − 3π)
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y(0) = 0, y 0 (0) = 0
MATH 250 Exam II-Sample3
9. (12 points) Find the inverse Laplace transform of the following functions F (s):
a)
F (s) =
e−s
s
b)
F (s) =
15se−7s
s2 + 9
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MATH 250 Exam II-Sample3
Answers: 1. False, False, True, True.
2. a) yp = (At + B)e−2t , b) yp = At2 + Bt + C, c) yp = Aet + t2 (Bt + C)e−t , d) yp = A cos t + B sin t
3. a) y = c1 e4t + c2 e−t − 2, b) y = c1 cos(5t) + c2 sin(5t) + (17/26)et
√
√
√
4. a) x00 +√x0 + 2x = √
0
x(0) = 0, x0 (0) = 1, b)x(t) =(2/ 7)e−t/2 sin( 7t/2), c)t = 0, t = 2π/ 7
d)t = (2/ 7)Arctan( 7),
√ √
√
√
√
√
e) x((2/ 7)Arctan( 7)) = (2/ 7)e−(Arctan 7/ 7) sin(Arctan 7)
5. a) (−∞, ∞), b)ck+2 =
−25ck
(k+2)(k+1)
valid from k = 0 to ∞.
6. The recursion relation is
ck+2 =
k(k + 1)ck+1 − ck
(k + 2)(k + 1)
which is valid for k ≥ 1, c2 = −c0 /2. And the 2 linearly indep. soln’s are
y1 = 1 −
x2 x3
−
+···
2
6
y2 = x −
x3 x4
−
+···
6
12
7. y = e−t cos t + e−t sin t + U(t − π)e−t+π sin(t − π)
8. y = (1/4)U(t − π) − (1/4)U(t − π) cos(2t − 2π) − (1/4)U(t − 3π) − (1/4)U(t − 3π) cos(2t − 6π)
9. a)U(t − 1), b) (15/3) cos(3t − 21).
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