In each of Problems 23 through 25, find the Wronskian of two solutions of the given differential equation without solving the equation. 23. 12y"-1(1+2) y' + (1+2)y=0 24. (cost) y" + (sin t) y' - ty = 0 25. (1-x²)y" - 2xy' + a(a + 1) y = 0, Legendre's equation
In each of Problems 23 through 25, find the Wronskian of two solutions of the given differential equation without solving the equation. 23. 12y"-1(1+2) y' + (1+2)y=0 24. (cost) y" + (sin t) y' - ty = 0 25. (1-x²)y" - 2xy' + a(a + 1) y = 0, Legendre's equation
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
25
![Problems
In each of Problems 1 through 5, find the Wronskian of the given pair
of functions.
1. e²r, e-3r/2
2. cost, sin t
3. e-21, te-21
4.
e' sint, e' cost
5. cos²0, 1+ cos(20)
In each of Problems 6 through 9, determine the longest interval in
which the given initial value problem is certain to have a unique twice-
differentiable solution. Do not attempt to find the solution.
6.
ty" + 3y = 1, y(1) = 1,
y'(1) = 2
7.
t(t-4)y" + 3ty' + 4y = 2,
y(3) = 0, y'(3) = -1
y" + (cost) y' + 3(ln |t) y = 0, y(2) = 3, y'(2) = 1
8.
9.
(x-2)y"+y+(x-2) (tan x) y = 0,
y(3) = 1, y'(3) = 2
10. Verify that y₁(t) 1² and y₂(1) = ¹ are two solutions
of the differential equation 12y" - 2y = 0 for t> 0. Then show
that y = c₁t² + c₂t is also a solution of this equation for any c₁
and C₂.
=
11. Verify that y₁ (1) = 1 and y₂(t) = 1/2 are solutions of the
differential equation yy" + (y')2 = 0 for t > 0. Then show that
y = C₁ + c₂t¹/² is not, in general, a solution of this equation. Explain
why this result does not contradict Theorem 3.2.2.
12. Show that if y = o(t) is a solution of the differential equation
y" + p(t) y' + q(t) y = g(t), where g(t) is not always zero, then
y = co(t), where c is any constant other than 1, is not a solution.
Explain why this result does not contradict the remark following
Theorem 3.2.2.
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 3e4t, 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.
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
Y4 = b1y1 + b2y2, where a₁, a2, b₁, and b2 are any constants. Show
that
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
In each of Problems 19 through 21, verify that the functions y₁ and y2
are solutions of the given differential equation. Do they constitute a
fundamental set of solutions?
19. y" +4y=0; y₁ (t) = cos(21), 32(1) = sin(21)
20. y" - 2y + y = 0; y₁ (t) = e', y2(1) = te'
21. x²y" - x(x+2) y' + (x + 2) y = 0, x > 0;
Ji(x) = x, yz(x) = xe*
22. Consider the equation y" - y' - 2y = 0.
a. Show that yı(t) = e' and y₂(t) = e2 form a fundamental
set of solutions.
b. Let y3(1)
-2e²t, ya(t)
= yı(t) + 2y2(1), and
ys(t) = 2y₁ (1)
2y₁ (1) - 2y3(1). Are y3(1), y4(1), and ys(t) also
solutions of the given differential equation?
c. Determine whether each of the following pairs forms a
fundamental set of solutions: {y₁(1), y(t)}; {12(1), 3(t)};
{yi(1), y4(1)}; {y4(1), ys(t)}.
In each of Problems 23 through 25, find the Wronskian of two
solutions of the given differential equation without solving the
equation.
23. 12y" -t(t+2) y' + (1+2) y = 0
24.
(cost) y" + (sin t) y' - ty = 0
25.
(1-x²) y" - 2xy' + a(a + 1) y = 0, Legendre's equation
26. Show that if p is differentiable and p(t) > 0, then the
Wronskian W(t) of two solutions of [p(t) y'] + q(t)y = 0 is
W(t) = c/p(t), where c is a constant.
27. If the differential equation ty" +2y' +te'y = 0 has y₁ and y2 as
a fundamental set of solutions and if W[y1, y2](1) = 2, find the value
of W[y1, y2](5).
28. If the Wronskian of any two solutions of y"+p(t) y'+q(t) y = 0
is constant, what does this imply about the coefficients p and q?
In Problems 29 and 30, assume that p and q are continuous and
that the functions y₁ and y2 are solutions of the differential equation
y" + p(t) y' +q(t) y = 0 on an open interval I.
29. Prove that if y₁ and y2 are zero at the same point in I, then they
cannot be a fundamental set of solutions on that interval.
30. Prove that if y, and y2 have a common point of inflection to in
I, then they cannot be a fundamental set of solutions on I unless both
p and q are zero at to.
31. Exact Equations. The equation
P(x) y" + Q(x) y' + R(x) y = 0
is said to be exact if it can be written in the form
(P(x) y')' + (f(x) y)' =
1
where f(x) is to be determined in terms of P(x), Q(x), and R(x).
The latter equation can be integrated once immediately, resulting
in a first-order linear equation for y that can be solved as in
Section 2.1. By equating the coefficients of the preceding equations
and then eliminating f(x), show that a necessary condition for
exactness is
P"(x) - Q'(x) + R(x) = 0.
1
It can be shown that this is also a sufficient condition.
In each of Problems 32 through 34, use the result of Problem 31 to
determine whether the given equation is exact. If it is, then solve the
equation.
32.
y" + xy + y = 0
33. xy" - (cos.x) y' + (sin x) y = 0, x > 0
34. x²y" + xy' - y = 0, x > 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f78378e-129e-42b7-9902-0b10b8fb85d5%2Ff5db23bb-e76b-4a6c-9468-cbacf56f05ca%2F8q3rklq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problems
In each of Problems 1 through 5, find the Wronskian of the given pair
of functions.
1. e²r, e-3r/2
2. cost, sin t
3. e-21, te-21
4.
e' sint, e' cost
5. cos²0, 1+ cos(20)
In each of Problems 6 through 9, determine the longest interval in
which the given initial value problem is certain to have a unique twice-
differentiable solution. Do not attempt to find the solution.
6.
ty" + 3y = 1, y(1) = 1,
y'(1) = 2
7.
t(t-4)y" + 3ty' + 4y = 2,
y(3) = 0, y'(3) = -1
y" + (cost) y' + 3(ln |t) y = 0, y(2) = 3, y'(2) = 1
8.
9.
(x-2)y"+y+(x-2) (tan x) y = 0,
y(3) = 1, y'(3) = 2
10. Verify that y₁(t) 1² and y₂(1) = ¹ are two solutions
of the differential equation 12y" - 2y = 0 for t> 0. Then show
that y = c₁t² + c₂t is also a solution of this equation for any c₁
and C₂.
=
11. Verify that y₁ (1) = 1 and y₂(t) = 1/2 are solutions of the
differential equation yy" + (y')2 = 0 for t > 0. Then show that
y = C₁ + c₂t¹/² is not, in general, a solution of this equation. Explain
why this result does not contradict Theorem 3.2.2.
12. Show that if y = o(t) is a solution of the differential equation
y" + p(t) y' + q(t) y = g(t), where g(t) is not always zero, then
y = co(t), where c is any constant other than 1, is not a solution.
Explain why this result does not contradict the remark following
Theorem 3.2.2.
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 3e4t, 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.
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
Y4 = b1y1 + b2y2, where a₁, a2, b₁, and b2 are any constants. Show
that
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
In each of Problems 19 through 21, verify that the functions y₁ and y2
are solutions of the given differential equation. Do they constitute a
fundamental set of solutions?
19. y" +4y=0; y₁ (t) = cos(21), 32(1) = sin(21)
20. y" - 2y + y = 0; y₁ (t) = e', y2(1) = te'
21. x²y" - x(x+2) y' + (x + 2) y = 0, x > 0;
Ji(x) = x, yz(x) = xe*
22. Consider the equation y" - y' - 2y = 0.
a. Show that yı(t) = e' and y₂(t) = e2 form a fundamental
set of solutions.
b. Let y3(1)
-2e²t, ya(t)
= yı(t) + 2y2(1), and
ys(t) = 2y₁ (1)
2y₁ (1) - 2y3(1). Are y3(1), y4(1), and ys(t) also
solutions of the given differential equation?
c. Determine whether each of the following pairs forms a
fundamental set of solutions: {y₁(1), y(t)}; {12(1), 3(t)};
{yi(1), y4(1)}; {y4(1), ys(t)}.
In each of Problems 23 through 25, find the Wronskian of two
solutions of the given differential equation without solving the
equation.
23. 12y" -t(t+2) y' + (1+2) y = 0
24.
(cost) y" + (sin t) y' - ty = 0
25.
(1-x²) y" - 2xy' + a(a + 1) y = 0, Legendre's equation
26. Show that if p is differentiable and p(t) > 0, then the
Wronskian W(t) of two solutions of [p(t) y'] + q(t)y = 0 is
W(t) = c/p(t), where c is a constant.
27. If the differential equation ty" +2y' +te'y = 0 has y₁ and y2 as
a fundamental set of solutions and if W[y1, y2](1) = 2, find the value
of W[y1, y2](5).
28. If the Wronskian of any two solutions of y"+p(t) y'+q(t) y = 0
is constant, what does this imply about the coefficients p and q?
In Problems 29 and 30, assume that p and q are continuous and
that the functions y₁ and y2 are solutions of the differential equation
y" + p(t) y' +q(t) y = 0 on an open interval I.
29. Prove that if y₁ and y2 are zero at the same point in I, then they
cannot be a fundamental set of solutions on that interval.
30. Prove that if y, and y2 have a common point of inflection to in
I, then they cannot be a fundamental set of solutions on I unless both
p and q are zero at to.
31. Exact Equations. The equation
P(x) y" + Q(x) y' + R(x) y = 0
is said to be exact if it can be written in the form
(P(x) y')' + (f(x) y)' =
1
where f(x) is to be determined in terms of P(x), Q(x), and R(x).
The latter equation can be integrated once immediately, resulting
in a first-order linear equation for y that can be solved as in
Section 2.1. By equating the coefficients of the preceding equations
and then eliminating f(x), show that a necessary condition for
exactness is
P"(x) - Q'(x) + R(x) = 0.
1
It can be shown that this is also a sufficient condition.
In each of Problems 32 through 34, use the result of Problem 31 to
determine whether the given equation is exact. If it is, then solve the
equation.
32.
y" + xy + y = 0
33. xy" - (cos.x) y' + (sin x) y = 0, x > 0
34. x²y" + xy' - y = 0, x > 0
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