Problems In each of Problems 1 through 8: Ga. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t→ ∞0. y' +3y=t+e-²1 1. 2. y' - 2y = 1²e²1 3. y' + y = te +1 4. y' +y = 3 cos(21), 1>0 5. y' - 2y = 3e' 6. ty' - y = 1²e-¹, 1>0 7. y' + y = 5 sin(21) 8. 2y + y = 31² In each of Problems 9 through 12, find the solution of the given initial value problem. Ove avloving og sk 9. y' - y = 2te²t, y(0) = 1 10. y' + 2y = te-2¹, y(1) = 0 2 cos t 11. y' +=y=₁2¹ y(a) = 0, t> 0 12. ty' + (t+1)y=t, y(ln 2) = 1, 1 > 0 In each of Problems 13 and 14: Ga. Draw a direction field for the given differential equation. How do solutions appear to behave as t becomes large? Does the behavior depend on the choice of the initial value a? Let ao be the value of a for which the transition from one type of behavior to another occurs. Estimate the value of ao. b. Solve the initial value problem and find the critical value ao exactly. c. Describe the behavior of the solution corresponding to the initial value ao. 13. yy=2 cost, 14. 3y - 2y = e-*1/2, and y(0) = a y(0) = a
Problems In each of Problems 1 through 8: Ga. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t→ ∞0. y' +3y=t+e-²1 1. 2. y' - 2y = 1²e²1 3. y' + y = te +1 4. y' +y = 3 cos(21), 1>0 5. y' - 2y = 3e' 6. ty' - y = 1²e-¹, 1>0 7. y' + y = 5 sin(21) 8. 2y + y = 31² In each of Problems 9 through 12, find the solution of the given initial value problem. Ove avloving og sk 9. y' - y = 2te²t, y(0) = 1 10. y' + 2y = te-2¹, y(1) = 0 2 cos t 11. y' +=y=₁2¹ y(a) = 0, t> 0 12. ty' + (t+1)y=t, y(ln 2) = 1, 1 > 0 In each of Problems 13 and 14: Ga. Draw a direction field for the given differential equation. How do solutions appear to behave as t becomes large? Does the behavior depend on the choice of the initial value a? Let ao be the value of a for which the transition from one type of behavior to another occurs. Estimate the value of ao. b. Solve the initial value problem and find the critical value ao exactly. c. Describe the behavior of the solution corresponding to the initial value ao. 13. yy=2 cost, 14. 3y - 2y = e-*1/2, and y(0) = a y(0) = a
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
9
Transcribed Image Text:Problems
In each of Problems 1 through 8:
Ga. Draw a direction field for the given differential equation.
b. Based on an inspection of the direction field, describe how
solutions behave for large t.
c. Find the general solution of the given differential equation,
and use it to determine how solutions behave as t→ ∞0.
y' +3y=t+e-²1
1.
2. y' - 2y = 1²e²1
3.
y' + y = te +1
4. y' + y = 3 cos(21), 1>0
5. y' - 2y = 3e'
6. ty'-y = 1²e-¹, 1>0
7. y' + y = 5 sin(21)
8. 2y + y = 31²
In each of Problems 9 through 12, find the solution of the given initial
value problem.
za vloving wong sw
9. y' - y = 2te²t,
10. y' + 2y = te-2¹,
2
cos t
11.
y' +=y=₁2¹
y(T) = 0, t> 0
12. ty' + (t+1)y=t, y(ln 2) = 1, t > 0
In each of Problems 13 and 14:
Ga. Draw a direction field for the given differential equation.
How do solutions appear to behave as t becomes large? Does the
behavior depend on the choice of the initial value a? Let a be
the value of a for which the transition from one type of behavior
to another occurs. Estimate the value of ao.
b. Solve the initial value problem and find the critical value ao
exactly.
c. Describe the behavior of the solution corresponding to the
initial value ao.
13.
yy=2 cost,
14. 3y - 2y = e-*1/2,
y(0) = 1
y(1) = 0
y(0) = a
y(0) = a
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