5. y"-6y' +9y=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
5
![can verify that the Wronskian of yı
TO OUX (TE)
(CE)
#0 for t> 0.
Upeni
tatt ball
Consequently, y, and y2 form a fundamental set of solutions of equation (33) for t > 0.
Problems
mont bonumine
In each of Problems 1 through 8, find the general solution of the given
differential equation.
10000
1. y" - 2y + y = 0
2. 9y" +6y' + y = 0
3.
4y" - 4y' - 3y = 0
4.
y" - 2y +10y = 0
5.
y" - 6y' +9y=0
6.
4y" +17y + 4y = 0
7. 16y" +24y' +9y = 0
8. 2y" +2y'+y=0
In each of Problems 9 through 11, solve the given initial value
problem. Sketch the graph of the solution and describe its behavior
for increasing t.
9. 9y" - 12y' + 4y = 0,
10. y"-6y' +9y = 0,
11. y" + 4y + 4y = 0,
y(0) = 2, y'(0) = -1
y(0) = 0, y'(0) = 2
y(-1) = 2, y'(-1) = 1
12. Consider the following modification of the initial value problem
in Example 2:
y" - y' += 0, y(0) = 2, y'(0) = b.
na
Find the solution as a function of b, and then determine the critical
value of b that separates solutions that remain positive for all t > 0
from those that eventually become negative.
3,-3/2
W[y₁, y2l(t) = 2
The Shosath
0=0x+1)+K
opb
N 13. Consider the initial value problem
4y" +4y' + y = 0, y(0) = 1, y'(0) = 2.
(36)
1600
a. Solve the initial value problem and plot the solution.
b. Determine the coordinates (tM, YM) of the maximum point.
c. Change the second initial condition to y'(0) = b > 0 and
find the solution as a function of b.
d. Find the coordinates (tM, YM) of the maximum point in terms
of b. Describe the dependence of tM and yM on b as b increases.
14. Consider the equation ay" + by' + cy= 0. If the roots of the
corresponding characteristic equation are real, show that a solution to
the differential equation either is everywhere zero or else can take on
the value zero at most once.
Problems 15 through 17 indicate other ways of finding the second
solution when the characteristic equation has repeated roots.
con
In each of
order to fir
15. a. Consider the equation y" + 2ay' + a²y = 0. Show that the
roots of the characteristic equation are r₁ r₂ = -a so that one
solution of the equation is eat.
b. Use Abel's formula [equation (23) of Section 3.2] to show
that the Wronskian of any two solutions of the given equation is
W(t) = y₁(t) y₂(t) - y₁ (1) y₂(t) = c₁e-2at,
18. 12y
19. Fy
20. 12
21. xy
where c₁ is a constant.
c. Let y₁ (t) =
eat and use the result of part b to obtain
a differential equation satisfied by a second solution y2(t). By
solving this equation, show that y2(t) = te-at.
22. X
23. T
arises
circu
and
24.
equ
sol](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fcedb1a-edf6-4a5a-b48b-980c99ebb17a%2Fa938445a-6536-4eee-8324-81d384687d61%2Fhb5cgbc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:can verify that the Wronskian of yı
TO OUX (TE)
(CE)
#0 for t> 0.
Upeni
tatt ball
Consequently, y, and y2 form a fundamental set of solutions of equation (33) for t > 0.
Problems
mont bonumine
In each of Problems 1 through 8, find the general solution of the given
differential equation.
10000
1. y" - 2y + y = 0
2. 9y" +6y' + y = 0
3.
4y" - 4y' - 3y = 0
4.
y" - 2y +10y = 0
5.
y" - 6y' +9y=0
6.
4y" +17y + 4y = 0
7. 16y" +24y' +9y = 0
8. 2y" +2y'+y=0
In each of Problems 9 through 11, solve the given initial value
problem. Sketch the graph of the solution and describe its behavior
for increasing t.
9. 9y" - 12y' + 4y = 0,
10. y"-6y' +9y = 0,
11. y" + 4y + 4y = 0,
y(0) = 2, y'(0) = -1
y(0) = 0, y'(0) = 2
y(-1) = 2, y'(-1) = 1
12. Consider the following modification of the initial value problem
in Example 2:
y" - y' += 0, y(0) = 2, y'(0) = b.
na
Find the solution as a function of b, and then determine the critical
value of b that separates solutions that remain positive for all t > 0
from those that eventually become negative.
3,-3/2
W[y₁, y2l(t) = 2
The Shosath
0=0x+1)+K
opb
N 13. Consider the initial value problem
4y" +4y' + y = 0, y(0) = 1, y'(0) = 2.
(36)
1600
a. Solve the initial value problem and plot the solution.
b. Determine the coordinates (tM, YM) of the maximum point.
c. Change the second initial condition to y'(0) = b > 0 and
find the solution as a function of b.
d. Find the coordinates (tM, YM) of the maximum point in terms
of b. Describe the dependence of tM and yM on b as b increases.
14. Consider the equation ay" + by' + cy= 0. If the roots of the
corresponding characteristic equation are real, show that a solution to
the differential equation either is everywhere zero or else can take on
the value zero at most once.
Problems 15 through 17 indicate other ways of finding the second
solution when the characteristic equation has repeated roots.
con
In each of
order to fir
15. a. Consider the equation y" + 2ay' + a²y = 0. Show that the
roots of the characteristic equation are r₁ r₂ = -a so that one
solution of the equation is eat.
b. Use Abel's formula [equation (23) of Section 3.2] to show
that the Wronskian of any two solutions of the given equation is
W(t) = y₁(t) y₂(t) - y₁ (1) y₂(t) = c₁e-2at,
18. 12y
19. Fy
20. 12
21. xy
where c₁ is a constant.
c. Let y₁ (t) =
eat and use the result of part b to obtain
a differential equation satisfied by a second solution y2(t). By
solving this equation, show that y2(t) = te-at.
22. X
23. T
arises
circu
and
24.
equ
sol
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