Download MATH 335: Applied Analysis I Sample Final Exam Set 1

Ivan Avramidi: (MATH 335: Applied Analysis I. Sample Final Exam)
MATH 335: Applied Analysis I
Sample Final Exam
Set 1
1. a) Find the solution of the initial value problem
2
cos t
y0 + y = 2 ,
t
t
y(π) = 0,
t>0
b) How does it behave as t → ∞?
2. a) Show that the differential equation
(3xy + y 2 )dx + (x2 + xy)dy = 0
is homogeneous.
b) Solve the differential equation.
3. a) Solve the initial value problem
2y 00 + 2y 0 + y = 0,
y(0) = 1,
y 0 (0) = 0 .
b) Sketch the graph of the solution and describe its behavior for increasing t.
4. Find the general solution of the differential equation
y 00 + y =
1
,
cos t
0<t<
5. Find the general solution of the differential equation
y (4) − 8y 0 = 0
Hint: Find all four roots of the characteristic equation.
6. Find the general solution of the differential equation
y (4) − y = 3t + et
π
.
2
1
Ivan Avramidi: (MATH 335: Applied Analysis I. Sample Final Exam)
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7. Solve the differential equation
y 00 − xy 0 − y = 0
by means of a power series about the point x0 = 0.
a) Find the recurrence relation.
b) Find the general term in each of two linearly independent solutions.
c) List the first four terms in each solution.
8. a) Show that the differential equation
x2 y 00 + xy 0 − 4x2 y = 0,
has a regular singular point at x = 0.
b) Determine the indicial equation.
c) Find the roots of indicial equation.
d) Find the recurrence relation.
e) Find one linearly independent series solution for x > 0 corresponding to one root (do
not find the second solution!).
9. Use the Laplace transform to solve the initial value problem
y 00 − 2y 0 + 2y = 0,
y(0) = 1,
y 0 (0) = 0.
10. Find the solution of the initial value problem
y 00 + 2y 0 + 2y = δ(t − π),
y(0) = 1,
y 0 (0) = 0 .
Ivan Avramidi: (MATH 335: Applied Analysis I. Sample Final Exam)
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Set 2
1. Section 2.1 a) Find the solution of the initial value problem
t3 y 0 + 4t2 y = e−t
y(−1) = 0 .
b) How does it behave as t → ∞?
2. Section 2.9 Solve the differential equation
dy
3y 2 − x2
=
.
dx
2x
3. Section 3.5 a) Solve the initial value problem
9y 00 + 6y 0 + 82y = 0,
y(0) = −1,
y 0 (0) = 2 .
b) Sketch the graph of the solution and describe its behavior for increasing t.
4. Section 3.7 Find the general solution of the differential equation
y 00 − 2y 0 + y =
et
t2 + 1
5. Section 4.2 Find the general solution of the differential equation
y 000 − 5y 00 + 3y 0 + y = 0
6. Section 4.3. Find the general solution of the differential equation
y (4) + 2y 00 + y = 3 + cos 2t
7. Section 5.2. Solve the differential equation by means of a power series about the
given point
x0 .
a) Find the recurrence relation.
b)Find the first four terms in each of two linearly independent solutions.
y 00 + xy 0 + 2y = 0,
x0 = 0
8. Section 5.6. a) Show that the differential equation has a regular singular point at
x = 0.
b) Determine the indicial equation c) the roots of indicial equation,
Ivan Avramidi: (MATH 335: Applied Analysis I. Sample Final Exam)
d) the recurrence relation.
e) Find the series solution for x > 0 corresponding to the larger root.
xy 00 + (1 − x)y 0 + λy = 0,
where λ is a constant.
9. Section 6.2. Use the Laplace transform to solve the initial value problem
y 00 + 2y 0 + y = 4e−t ,
y(0) = 2, y 0 (0) = −1.
10. Section 6.5 Find the solution of the initial value problem
y 00 + 2y 0 + 2y = cost + δ(t − π/2),
y(0) = 0, y 0 (0) = 0
4
Ivan Avramidi: (MATH 335: Applied Analysis I. Sample Final Exam)
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Set 3
1. (a) Find the solution of the initial value problem
t2 y 0 + 2ty = sin t,
y(π) = 0,
t>0
(b) How does it behave as t → ∞?
2. (a) Show that the differential equation
2xy dx + (3y 2 − x2 ) dy = 0
is homogeneous.
(b) Solve the differential equation.
3. (a) Solve the initial value problem
y 00 + 4y 0 + 4y = 0,
y 0 (−1) = 1 .
y(−1) = 2,
(b) Sketch the graph of the solution and describe its behavior for increasing t.
4. Find the general solution of the differential equation
y 00 + 4y 0 + 4y =
e−2t
,
t2
t > 0.
5. Find the general solution of the differential equation
y (4) − 5y 00 + 4y = 0
6. Find the general solution of the differential equation
y (4) + y 000 = sin(2t)
7. Solve the differential equation
y 00 + x2 y = 0
by means of a power series about the point x0 = 0.
(a) Find the recurrence relation. (b) Find the first two nonvanishing terms in each of
two linearly independent solutions. (c) If possible, find the general term in each solution.
Ivan Avramidi: (MATH 335: Applied Analysis I. Sample Final Exam)
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8. (a) Show that the differential equation
1
2 00
0
2
x y + xy + x −
y = 0,
4
has a regular singular point at x = 0. (b) Determine the indicial equation. (c) Find
the roots of indicial equation. (d) Find the recurrence relation. (e) Find two linearly
independent series solution for x > 0 (list two nonvanishing terms in each solution). (f )
If possible, find find the general term in each solution, and then find expressions for the
solutions in terms of elementary functions. Hint: The linearly independent solutions are
sin x
cos x
√ and √ .
x
x
9. Use the Laplace transform to solve the initial value problem
y 00 + 2y 0 + 5y = 0,
y(0) = 2,
y 0 (0) = −1 .
10. Find the solution of the initial value problem
y 00 + 4y = δ(t − 4π),
1
y(0) = ,
2
y 0 (0) = 0 .