29. a. If a > 0 and c> 0, but b = 0, show that the result of Problem 28 is no longer true, but that all solutions are bounded as to. b. If a > 0 and b > 0, but c = 0, show that the result of Problem 28 is no longer true, but that all solutions approach a constant that depends on the initial conditions as too. Determine this constant for the initial conditions y(0) = yo, y'(0) = yo
29. a. If a > 0 and c> 0, but b = 0, show that the result of Problem 28 is no longer true, but that all solutions are bounded as to. b. If a > 0 and b > 0, but c = 0, show that the result of Problem 28 is no longer true, but that all solutions approach a constant that depends on the initial conditions as too. Determine this constant for the initial conditions y(0) = yo, y'(0) = yo
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
29
![+ br + c = 0 and that
lutions of the differential
71²
1
Then think of r₁ as fixed,
fo(t; r₁, r₂) as r2 → r1,
case of equal roots.
r1, show that
= a(r-r₁)²e²¹.
(37)
7) is zero when r = r₁, it
]=ay"+by'+cy = 0.
pect to r, and interchange
with respect to t, thus
te"]
2ae¹¹ (r-r₁).
(38)
6) is zero when r = r₁,
n of L[y] = 0.
method of reduction of
fferential equation.
= 1²
= t
=t-1
Inou
= sin(x²)
y₁(x) = x-1/2 sin x
X
uniform stream past a
5x2/2) is one solution,
of an integral.
cended to second-order
a known nonvanishing
hat a second solution y2
satisfies (y2/y₁)' = W[y1, y2l/y₁, where W[y₁, y2] is the Wronskian
of y₁ and y2. Then use Abel's formula (equation (23) of Section 3.2)
to determine y2.
In each of Problems 25 through 27, use the method of Problem 24 to
find a second independent solution of the given equation.
25. t²y" + 3ty' + y = 0, t> 0;
26. ty" - y' + 4t³y = 0, t> 0;
27. x²y"+xy'+(x² -0.25) y = 0,
y₁(t) = t-¹
y₁(t) = sin(t²)
x > 0; y₁(x) = x-¹/2 sin x
Behavior of Solutions as t→∞. Problems 28 through 30 are
concerned with the behavior of solutions as too.
28. If a, b, and c are positive constants, show that all solutions of
ay" +by' + cy = 0 approach zero as t → ∞.
29. a. If a > 0 and c> 0, but b = 0, show that the result of
Problem 28 is no longer true, but that all solutions are bounded
as t.
b. If a > 0 and b > 0, but c = 0, show that the result of
Problem 28 is no longer true, but that all solutions
approach a constant that depends on the initial conditions as
t → ∞. Determine this constant for the initial conditions
y(0) = yo, y'(0) = yo.
30. Show that y = sint is a solution of
y" + (k sin² t) y' + (1 - k cost sin t) y = 0
for any value of the constant k. If 0 <k < 2, show that
1 - k cost sint > 0 and k sin² t≥ 0. Thus observe that even though
the coefficients of this variable-coefficient differential equation are
nonnegative (and the coefficient of y' is zero only at the points
t = 0, π, 2π, ...), it has a solution that does not approach zero
as t->> ∞. Compare this situation with the result of Problem 28.
Thus we observe a not unusual situation in the study of differential
equations: equations that are apparently very similar can have quite
different properties.
Euler Equations. In each of Problems 31 through 34, use the
substitution introduced in Problem 25 in Section 3.3 to solve the given
differential equation.
31. ty" - 3ty' + 4y = 0, t> 0
32.
33.
t²y" +2ty' +0.25y = 0, t> 0
t²y" + 3ty' + y = 0, t> 0
34. 4t2y" - 8ty' +9y = 0, t> 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1be8149a-7f70-4e17-bee2-140ac0078ae6%2Ffab4893c-afee-4fea-8a0a-6aaee47e8034%2Frxrxvlo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:+ br + c = 0 and that
lutions of the differential
71²
1
Then think of r₁ as fixed,
fo(t; r₁, r₂) as r2 → r1,
case of equal roots.
r1, show that
= a(r-r₁)²e²¹.
(37)
7) is zero when r = r₁, it
]=ay"+by'+cy = 0.
pect to r, and interchange
with respect to t, thus
te"]
2ae¹¹ (r-r₁).
(38)
6) is zero when r = r₁,
n of L[y] = 0.
method of reduction of
fferential equation.
= 1²
= t
=t-1
Inou
= sin(x²)
y₁(x) = x-1/2 sin x
X
uniform stream past a
5x2/2) is one solution,
of an integral.
cended to second-order
a known nonvanishing
hat a second solution y2
satisfies (y2/y₁)' = W[y1, y2l/y₁, where W[y₁, y2] is the Wronskian
of y₁ and y2. Then use Abel's formula (equation (23) of Section 3.2)
to determine y2.
In each of Problems 25 through 27, use the method of Problem 24 to
find a second independent solution of the given equation.
25. t²y" + 3ty' + y = 0, t> 0;
26. ty" - y' + 4t³y = 0, t> 0;
27. x²y"+xy'+(x² -0.25) y = 0,
y₁(t) = t-¹
y₁(t) = sin(t²)
x > 0; y₁(x) = x-¹/2 sin x
Behavior of Solutions as t→∞. Problems 28 through 30 are
concerned with the behavior of solutions as too.
28. If a, b, and c are positive constants, show that all solutions of
ay" +by' + cy = 0 approach zero as t → ∞.
29. a. If a > 0 and c> 0, but b = 0, show that the result of
Problem 28 is no longer true, but that all solutions are bounded
as t.
b. If a > 0 and b > 0, but c = 0, show that the result of
Problem 28 is no longer true, but that all solutions
approach a constant that depends on the initial conditions as
t → ∞. Determine this constant for the initial conditions
y(0) = yo, y'(0) = yo.
30. Show that y = sint is a solution of
y" + (k sin² t) y' + (1 - k cost sin t) y = 0
for any value of the constant k. If 0 <k < 2, show that
1 - k cost sint > 0 and k sin² t≥ 0. Thus observe that even though
the coefficients of this variable-coefficient differential equation are
nonnegative (and the coefficient of y' is zero only at the points
t = 0, π, 2π, ...), it has a solution that does not approach zero
as t->> ∞. Compare this situation with the result of Problem 28.
Thus we observe a not unusual situation in the study of differential
equations: equations that are apparently very similar can have quite
different properties.
Euler Equations. In each of Problems 31 through 34, use the
substitution introduced in Problem 25 in Section 3.3 to solve the given
differential equation.
31. ty" - 3ty' + 4y = 0, t> 0
32.
33.
t²y" +2ty' +0.25y = 0, t> 0
t²y" + 3ty' + y = 0, t> 0
34. 4t2y" - 8ty' +9y = 0, t> 0
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