In each of Problems 17 and 18, find the fundamental set of solutions specified by Theorem 3.2.5 for the given differential equation and initial point. 17. y"+y' - 2y = 0, to = 0
In each of Problems 17 and 18, find the fundamental set of solutions specified by Theorem 3.2.5 for the given differential equation and initial point. 17. y"+y' - 2y = 0, to = 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
17
![erve
this
= 0
2).
if
al
Since W#0 for t > 0, we conclude that y₁ and y2 form a fundamental set of solutions there. Thus
the general solution of differential equation (14) is y(t) = c₁t¹/2 + c₂t¹ fort > 0.
In several cases we have been able to find a fundamental set of solutions, and therefore the
general solution, of a given differential equation. However, this is often a difficult task, and the
question arises as to whether a differential equation of the form (2) always has a fundamental
set of solutions. The following theorem provides an affirmative answer to this question.
Theorem 3.2.5
Consider the differential equation (2),
L[y] =y" + p(t) y' +q(t)y=0,
whose coefficients p and q are continuous on some open interval I. Choose some point to in I. Let
y be the solution of equation (2) that also satisfies the initial conditions
y(to) = 1, y'(to) = 0,
and let y2 be the solution of equation (2) that satisfies the initial conditions
y(to) = 0, y'(to) = 1.
Then y, and y2 form a fundamental set of solutions of equation (2).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f78378e-129e-42b7-9902-0b10b8fb85d5%2F1460e2dc-37df-4561-88bf-7d937cc281cb%2F0qq2ev8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:erve
this
= 0
2).
if
al
Since W#0 for t > 0, we conclude that y₁ and y2 form a fundamental set of solutions there. Thus
the general solution of differential equation (14) is y(t) = c₁t¹/2 + c₂t¹ fort > 0.
In several cases we have been able to find a fundamental set of solutions, and therefore the
general solution, of a given differential equation. However, this is often a difficult task, and the
question arises as to whether a differential equation of the form (2) always has a fundamental
set of solutions. The following theorem provides an affirmative answer to this question.
Theorem 3.2.5
Consider the differential equation (2),
L[y] =y" + p(t) y' +q(t)y=0,
whose coefficients p and q are continuous on some open interval I. Choose some point to in I. Let
y be the solution of equation (2) that also satisfies the initial conditions
y(to) = 1, y'(to) = 0,
and let y2 be the solution of equation (2) that satisfies the initial conditions
y(to) = 0, y'(to) = 1.
Then y, and y2 form a fundamental set of solutions of equation (2).
![13. Can y = sin(1²) be a solution on an interval containing t = 0 of
an equation y" + p(t) y' + q(t) y = 0 with continuous coefficients?
Explain your answer.
14. If the Wronskian W of f and g is 3e4, and if f(t) = e²t, find
g(t).
15. If the Wronskian of f and g is t cost - sint, and if
u = f + 3g, v = f - g, find the Wronskian of u and v.
mis
16. Assume that y₁ and y2 are a fundamental set of solutions
of y" + p(t) y' + q(t) y
0 and let y3 = a₁y₁ + a2y2 and
bıyı + b2y2, where a₁, a2, b₁, and b2 are any constants. Show
=
=
Y4
that
29.
cann
30.
19. y" +4y=0; y₁ (t) = cos(2t), y2(t) = sin(2t)
20. y" - 2y' + y = 0; y₁ (t) = e',
y2(t) = te'
21. x²y" - x(x+2) y' + (x + 2) y = 0, x > 0;
Ji(x) = x, yz(x) = xe*
R
I, th
pa
31.
W[y3, y4] = (a₁b₂-a2b₁) W[y₁, y2].
Are y3 and y4 also a fundamental set of solutions? Why or why not?
In each of Problems 17 and 18, find the fundamental set of solutions
specified by Theorem 3.2.5 for the given differential equation and
initial point.
17. y"+y' - 2y = 0, to = 0
18. y" +4y' + 3y = 0,
to = 1
Inithanogs
In each of Problems 19 through 21, verify that the functions y, and y2
are solutions of the given differential equation. Do they constitute a
fundamental set of solutions?
V](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f78378e-129e-42b7-9902-0b10b8fb85d5%2F1460e2dc-37df-4561-88bf-7d937cc281cb%2Fytxuczq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:13. Can y = sin(1²) be a solution on an interval containing t = 0 of
an equation y" + p(t) y' + q(t) y = 0 with continuous coefficients?
Explain your answer.
14. If the Wronskian W of f and g is 3e4, and if f(t) = e²t, find
g(t).
15. If the Wronskian of f and g is t cost - sint, and if
u = f + 3g, v = f - g, find the Wronskian of u and v.
mis
16. Assume that y₁ and y2 are a fundamental set of solutions
of y" + p(t) y' + q(t) y
0 and let y3 = a₁y₁ + a2y2 and
bıyı + b2y2, where a₁, a2, b₁, and b2 are any constants. Show
=
=
Y4
that
29.
cann
30.
19. y" +4y=0; y₁ (t) = cos(2t), y2(t) = sin(2t)
20. y" - 2y' + y = 0; y₁ (t) = e',
y2(t) = te'
21. x²y" - x(x+2) y' + (x + 2) y = 0, x > 0;
Ji(x) = x, yz(x) = xe*
R
I, th
pa
31.
W[y3, y4] = (a₁b₂-a2b₁) W[y₁, y2].
Are y3 and y4 also a fundamental set of solutions? Why or why not?
In each of Problems 17 and 18, find the fundamental set of solutions
specified by Theorem 3.2.5 for the given differential equation and
initial point.
17. y"+y' - 2y = 0, to = 0
18. y" +4y' + 3y = 0,
to = 1
Inithanogs
In each of Problems 19 through 21, verify that the functions y, and y2
are solutions of the given differential equation. Do they constitute a
fundamental set of solutions?
V
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