126 CHAPTER 3 Second-Order Linear Differential Equations 19. Show that W[et cos(ut), et sin(ur)] = μe²¹. 20. In this problem we outline a different derivation of Euler's formula. a. Show that y₁ (1) = cost and y₂(1) = sint are a fundamental set of solutions of y"+y = 0; that is, show that they are solutions and that their Wronskian is not zero. b. Show (formally) that y = et is also a solution of y" + y = 0. Therefore, eit = C₁ cost + C₂ sin t for some constants c₁ and c₂. Why is this so? c. Set = 0 in equation (31) to show that c₁ = 1. d. Assuming that equation (15) is true, differentiate equation (31) and then set t = 0 to conclude that c₂ = i. Use the values of c₁ and c₂ in equation (31) to arrive at Euler's formula. 21. Using Euler's formula, show that eit + e-it 2 eit - e-it 2i = cost, 22. If et is given by equation (14), show that e(+2)² = e'1¹e²2² for any complex numbers r₁ and 72. 23. Consider the differential equation = sint. ay" +by+cy = 0, (31) where b²-4ac < 0 and the characteristic equation has complex roots Aiu. Substitute the functions u(t)= e cos(ut) and v(t) = e sin(pt) 12 for y in the differential equation and thereby confirm that they are solutions. d²y d1² 24. If the functions y₁ and y2 are a fundamental set of solutions of y"+p(t)y'+q(t) y = 0, show that between consecutive zeros of y₁ there is one and only one zero of y2. Note that this result is illustrated by the solutions y₁ (t) = cost and y2(1) = sint of the equation y" + y = 0. Hint: Suppose that 1₁ and 12 are two zeros of y₁ between which there are no zeros of y2. Apply Rolle's theorem to y₁/2 to reach a contradiction. Change of Variables. Sometimes a differential equation with variable coefficients, y" + p(t)y' +q(t) y = 0, (32) can be put in a more suitable form for finding a solution by making a change of the independent variable. We explore these ideas in Problems 25 through 36. In particular, in Problem 25 we show that a class of equations known as Euler equations can be transformed into equations with constant coefficients by a simple change of the independent variable. Problems 26 through 31 are examples of this type of equation. Problem 32 determines conditions under which the more general equation (32) can be transformed into a differential equation with constant coefficients. Problems 33 through 36 give specific applications of this procedure. 25. Euler Equations. An equation of the form dy + at +By=0, t > 0, dt (33) where a and 3 are real constants, is called an Euler equation. a. Let x = Int and calculate dy/dt and d²y/di² in terms of dy/dx and d²y/dx². b. Use the results of part a to transform equation (33) into dy +(a - 1) / d² dx2 (34) Observe that differential equation (34) has constant coefficients. If y₁(x) and y₂(x) form a fundamental set of solutions of of solutions of equation (33). equation (34), then y₁ (Int) and y₂(Int) form a fundamental set In each of Problems 26 through 31, use the method of Problem 25 to solve the given equation for > 0. 26. 12y" +ty' + y = 0 27. 12y" +4ty' + 2y = 0 28. 12y"-4ty' - 6y = 0 29. 12y"-4ty' +6y=0 30. 12y" + 3ty' - 3y = 0 31. 12y" +7ty' + 10y = 0 32. In this problem we determine conditions on p and q that enable equation (32) to be transformed into an equation with constant coefficients by a change of the independent variable. Let x = u(1) be the new independent variable, with the relation between x and t to be specified later. a. Show that dy dt dx d²y (4) 4 b. Show that the differential equation (32) becomes ²d²y dx² d²x dt² = dx dt dx dy dt dx' d'y dt² + +By = 0. = dx dy dt dx c. In order for equation (35) to have constant coefficients, the coefficients of d²y/dx², dy/dx, and y must all be proportional. If q(t) > 0, then we can choose the constant of proportionality to be 1; hence, after integrating with respect to t, + p(t) x = u(t) = d²x dy + dt dx² dt2 dx +q(t)y=0. (35) = f(g(x))¹/2 dr. ¹/² dt. Sido (36) vid. With x chosen as in part c, show that the coefficient of dy/dx in equation (35) is also a constant, provided that the expression q'(t) +2p(t)q(t) 2(q(t))³/2 (37) is a constant. Thus equation (32) can be transformed into an equation with constant coefficients by a change of the independent variable, provided that the function (q' +2pq)/9³/2 is a constant. e. How must the analysis and results in d be modified if q(1) < 0? In each of Problems 33 through 36, try to transform the given equation into one with constant coefficients by the method of Problem 32. If this is possible, find the general solution of the given equation. 33. y" +ty'+e-¹² y = 0, -∞ < t < 8 188 34. y" + 3ty' +1²y = 0, 35. ty" +(12-1) y' + ³y = 0, 0
126 CHAPTER 3 Second-Order Linear Differential Equations 19. Show that W[et cos(ut), et sin(ur)] = μe²¹. 20. In this problem we outline a different derivation of Euler's formula. a. Show that y₁ (1) = cost and y₂(1) = sint are a fundamental set of solutions of y"+y = 0; that is, show that they are solutions and that their Wronskian is not zero. b. Show (formally) that y = et is also a solution of y" + y = 0. Therefore, eit = C₁ cost + C₂ sin t for some constants c₁ and c₂. Why is this so? c. Set = 0 in equation (31) to show that c₁ = 1. d. Assuming that equation (15) is true, differentiate equation (31) and then set t = 0 to conclude that c₂ = i. Use the values of c₁ and c₂ in equation (31) to arrive at Euler's formula. 21. Using Euler's formula, show that eit + e-it 2 eit - e-it 2i = cost, 22. If et is given by equation (14), show that e(+2)² = e'1¹e²2² for any complex numbers r₁ and 72. 23. Consider the differential equation = sint. ay" +by+cy = 0, (31) where b²-4ac < 0 and the characteristic equation has complex roots Aiu. Substitute the functions u(t)= e cos(ut) and v(t) = e sin(pt) 12 for y in the differential equation and thereby confirm that they are solutions. d²y d1² 24. If the functions y₁ and y2 are a fundamental set of solutions of y"+p(t)y'+q(t) y = 0, show that between consecutive zeros of y₁ there is one and only one zero of y2. Note that this result is illustrated by the solutions y₁ (t) = cost and y2(1) = sint of the equation y" + y = 0. Hint: Suppose that 1₁ and 12 are two zeros of y₁ between which there are no zeros of y2. Apply Rolle's theorem to y₁/2 to reach a contradiction. Change of Variables. Sometimes a differential equation with variable coefficients, y" + p(t)y' +q(t) y = 0, (32) can be put in a more suitable form for finding a solution by making a change of the independent variable. We explore these ideas in Problems 25 through 36. In particular, in Problem 25 we show that a class of equations known as Euler equations can be transformed into equations with constant coefficients by a simple change of the independent variable. Problems 26 through 31 are examples of this type of equation. Problem 32 determines conditions under which the more general equation (32) can be transformed into a differential equation with constant coefficients. Problems 33 through 36 give specific applications of this procedure. 25. Euler Equations. An equation of the form dy + at +By=0, t > 0, dt (33) where a and 3 are real constants, is called an Euler equation. a. Let x = Int and calculate dy/dt and d²y/di² in terms of dy/dx and d²y/dx². b. Use the results of part a to transform equation (33) into dy +(a - 1) / d² dx2 (34) Observe that differential equation (34) has constant coefficients. If y₁(x) and y₂(x) form a fundamental set of solutions of of solutions of equation (33). equation (34), then y₁ (Int) and y₂(Int) form a fundamental set In each of Problems 26 through 31, use the method of Problem 25 to solve the given equation for > 0. 26. 12y" +ty' + y = 0 27. 12y" +4ty' + 2y = 0 28. 12y"-4ty' - 6y = 0 29. 12y"-4ty' +6y=0 30. 12y" + 3ty' - 3y = 0 31. 12y" +7ty' + 10y = 0 32. In this problem we determine conditions on p and q that enable equation (32) to be transformed into an equation with constant coefficients by a change of the independent variable. Let x = u(1) be the new independent variable, with the relation between x and t to be specified later. a. Show that dy dt dx d²y (4) 4 b. Show that the differential equation (32) becomes ²d²y dx² d²x dt² = dx dt dx dy dt dx' d'y dt² + +By = 0. = dx dy dt dx c. In order for equation (35) to have constant coefficients, the coefficients of d²y/dx², dy/dx, and y must all be proportional. If q(t) > 0, then we can choose the constant of proportionality to be 1; hence, after integrating with respect to t, + p(t) x = u(t) = d²x dy + dt dx² dt2 dx +q(t)y=0. (35) = f(g(x))¹/2 dr. ¹/² dt. Sido (36) vid. With x chosen as in part c, show that the coefficient of dy/dx in equation (35) is also a constant, provided that the expression q'(t) +2p(t)q(t) 2(q(t))³/2 (37) is a constant. Thus equation (32) can be transformed into an equation with constant coefficients by a change of the independent variable, provided that the function (q' +2pq)/9³/2 is a constant. e. How must the analysis and results in d be modified if q(1) < 0? In each of Problems 33 through 36, try to transform the given equation into one with constant coefficients by the method of Problem 32. If this is possible, find the general solution of the given equation. 33. y" +ty'+e-¹² y = 0, -∞ < t < 8 188 34. y" + 3ty' +1²y = 0, 35. ty" +(12-1) y' + ³y = 0, 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
29,33
![126 CHAPTER 3 Second-Order Linear Differential Equations
19. Show that W [e cos(μt), e sin(pt)] = μe²¹.
20. In this problem we outline a different derivation of Euler's
formula.
a. Show that y₁ (1) = cost and y₂ (1) = sint are a fundamental
set of solutions of y" + y = 0; that is, show that they are solutions
and that their Wronskian is not zero.
b. Show (formally) that y = eit is also a solution of y" + y = 0.t
Therefore,
eit = c₁ cost + c₂ sin t
for some constants c₁ and c₂. Why is this so?
c. Set t = 0 in equation (31) to show that c₁ = 1.
d. Assuming that equation (15) is true, differentiate
equation (31) and then set t = 0 to conclude that c₂ = i. Use the
values of c₁ and c2 in equation (31) to arrive at Euler's formula.
21. Using Euler's formula, show that
eit + e-it
2
eit - e-it
2i
= cost,
= sin t.
22. If et is given by equation (14), show that e(+2)t = e'1¹e²2¹
for any complex numbers r₁ and 12.
23. Consider the differential equation
ay" +by' + cy = 0,
(31)
where b²-4ac < 0 and the characteristic equation has complex roots
Aiu. Substitute the functions
37947
u(t) = et cos(ut) and v(t) = et sin(µt)
for y in the differential equation and thereby confirm that they are
solutions.
120²
dt²
=
24. If the functions y₁ and y2 are a fundamental set of solutions of
y" +p(t)y'+q(t) y = 0, show that between consecutive zeros of y₁
there is one and only one zero of y2. Note that this result is illustrated
by the solutions yı(t) : cost and y₂(1) = sint of the equation
y" + y = 0.
Hint: Suppose that t₁ and 2 are two zeros of y₁ between which
there are no zeros of y2. Apply Rolle's theorem to y₁/y2 to reach a
contradiction.
Change of Variables. Sometimes a differential equation with variable
coefficients,
y" + p(t)y' + g(t) y = 0,
(32)
can be put in a more suitable form for finding a solution by making
a change of the independent variable. We explore these ideas in
Problems 25 through 36. In particular, in Problem 25 we show that
a class of equations known as Euler equations can be transformed
into equations with constant coefficients by a simple change of the
independent variable. Problems 26 through 31 are examples of this
type of equation. Problem 32 determines conditions under which the
more general equation (32) can be transformed into a differential
equation with constant coefficients. Problems 33 through 36 give
specific applications of this procedure.
25. Euler Equations. An equation of the form
dy
+at- +By=0, t > 0,
dt
(33)
where a and 3 are real constants, is called an Euler equation.
a. Let x = lnt and calculate dy/dt and d²y/dt² in terms of
dy/dx and d2y/dx².
b. Use the results of part a to transform equation (33) into
d²y
dx²
dy
+(a-1) +
dx
Observe that differential equation (34) has constant coefficients.
If y₁(x) and y2(x) form a fundamental set of solutions of
of solutions of equation (33).
equation (34), then y₁ (Int) and y2(Int) form a fundamental set
clareza
In each of Problems 26 through 31, use the method of Problem 25 to
solve the given equation for t > 0.
26. 1²y" +ty' + y = 0
27. 12y" +4ty' + 2y = 0
28. t2y" - 4ty' - 6y=0
29. t2y"-4ty' +6y=0
30. 12y" + 3ty' - 3y = 0
31. 12y" +7ty' +10y = 0.
32. In this problem we determine conditions on p and q that
enable equation (32) to be transformed into an equation with constant
coefficients by a change of the independent variable. Let x = u(t) be
the new independent variable, with the relation between x and t to be
specified later.
=
a. Show that
dy dx dy
dt
=
dt dx'
d²y
dt²
2
(4) + (4
dx d²y d²x
dt
dx²
+ By = 0.
dx
- (+) ²
=
dt
+ p(t)
dt²
dx
dt
Hig
b. Show that the differential equation (32) becomes
:)
d²y
dx²
dy
+
dx
(34)
W
d²x dy
dt² dx
x = u(t) = f(q(1))}¹/² dt.
+q(t) y = 0. (35)
c. In order for equation (35) to have constant coefficients, the
coefficients of d²y/dx2, dy/dx, and y must all be proportional.
If q(t) > 0, then we can choose the constant of proportionality
to be 1; hence, after integrating with respect to t,
dt. eldo (36)
navid. With x chosen as in part c, show that the coefficient of dy/dx
in equation (35) is also a constant, provided that the expression
q'(t) +2p(t)q(t)
2(q(t))³/2
(37)
is a constant. Thus equation (32) can be transformed into
an equation with constant coefficients by a change of the
independent variable, provided that the function (q'+2pq)/9³/2
is a constant.
e. How must the analysis and results in d be modified if
q(t) < 0?
In each of Problems 33 through 36, try to transform the given equation
into one with constant coefficients by the method of Problem 32. If this
is possible, find the general solution of the given equation.
33. y" +ty'+e-1² y = 0,
188
34. y" + 3ty' +1²y = 0,
-∞ < t < ∞
35. ty" + (t²-1)y' + ³y = 0, 0< t << ∞
36. y" +ty' - e-t²y = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e88b07-245a-4a34-a7e7-6bf169a590be%2F2aafd8c2-28bc-4c0d-80db-edafcd518a92%2F1v9xrfn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:126 CHAPTER 3 Second-Order Linear Differential Equations
19. Show that W [e cos(μt), e sin(pt)] = μe²¹.
20. In this problem we outline a different derivation of Euler's
formula.
a. Show that y₁ (1) = cost and y₂ (1) = sint are a fundamental
set of solutions of y" + y = 0; that is, show that they are solutions
and that their Wronskian is not zero.
b. Show (formally) that y = eit is also a solution of y" + y = 0.t
Therefore,
eit = c₁ cost + c₂ sin t
for some constants c₁ and c₂. Why is this so?
c. Set t = 0 in equation (31) to show that c₁ = 1.
d. Assuming that equation (15) is true, differentiate
equation (31) and then set t = 0 to conclude that c₂ = i. Use the
values of c₁ and c2 in equation (31) to arrive at Euler's formula.
21. Using Euler's formula, show that
eit + e-it
2
eit - e-it
2i
= cost,
= sin t.
22. If et is given by equation (14), show that e(+2)t = e'1¹e²2¹
for any complex numbers r₁ and 12.
23. Consider the differential equation
ay" +by' + cy = 0,
(31)
where b²-4ac < 0 and the characteristic equation has complex roots
Aiu. Substitute the functions
37947
u(t) = et cos(ut) and v(t) = et sin(µt)
for y in the differential equation and thereby confirm that they are
solutions.
120²
dt²
=
24. If the functions y₁ and y2 are a fundamental set of solutions of
y" +p(t)y'+q(t) y = 0, show that between consecutive zeros of y₁
there is one and only one zero of y2. Note that this result is illustrated
by the solutions yı(t) : cost and y₂(1) = sint of the equation
y" + y = 0.
Hint: Suppose that t₁ and 2 are two zeros of y₁ between which
there are no zeros of y2. Apply Rolle's theorem to y₁/y2 to reach a
contradiction.
Change of Variables. Sometimes a differential equation with variable
coefficients,
y" + p(t)y' + g(t) y = 0,
(32)
can be put in a more suitable form for finding a solution by making
a change of the independent variable. We explore these ideas in
Problems 25 through 36. In particular, in Problem 25 we show that
a class of equations known as Euler equations can be transformed
into equations with constant coefficients by a simple change of the
independent variable. Problems 26 through 31 are examples of this
type of equation. Problem 32 determines conditions under which the
more general equation (32) can be transformed into a differential
equation with constant coefficients. Problems 33 through 36 give
specific applications of this procedure.
25. Euler Equations. An equation of the form
dy
+at- +By=0, t > 0,
dt
(33)
where a and 3 are real constants, is called an Euler equation.
a. Let x = lnt and calculate dy/dt and d²y/dt² in terms of
dy/dx and d2y/dx².
b. Use the results of part a to transform equation (33) into
d²y
dx²
dy
+(a-1) +
dx
Observe that differential equation (34) has constant coefficients.
If y₁(x) and y2(x) form a fundamental set of solutions of
of solutions of equation (33).
equation (34), then y₁ (Int) and y2(Int) form a fundamental set
clareza
In each of Problems 26 through 31, use the method of Problem 25 to
solve the given equation for t > 0.
26. 1²y" +ty' + y = 0
27. 12y" +4ty' + 2y = 0
28. t2y" - 4ty' - 6y=0
29. t2y"-4ty' +6y=0
30. 12y" + 3ty' - 3y = 0
31. 12y" +7ty' +10y = 0.
32. In this problem we determine conditions on p and q that
enable equation (32) to be transformed into an equation with constant
coefficients by a change of the independent variable. Let x = u(t) be
the new independent variable, with the relation between x and t to be
specified later.
=
a. Show that
dy dx dy
dt
=
dt dx'
d²y
dt²
2
(4) + (4
dx d²y d²x
dt
dx²
+ By = 0.
dx
- (+) ²
=
dt
+ p(t)
dt²
dx
dt
Hig
b. Show that the differential equation (32) becomes
:)
d²y
dx²
dy
+
dx
(34)
W
d²x dy
dt² dx
x = u(t) = f(q(1))}¹/² dt.
+q(t) y = 0. (35)
c. In order for equation (35) to have constant coefficients, the
coefficients of d²y/dx2, dy/dx, and y must all be proportional.
If q(t) > 0, then we can choose the constant of proportionality
to be 1; hence, after integrating with respect to t,
dt. eldo (36)
navid. With x chosen as in part c, show that the coefficient of dy/dx
in equation (35) is also a constant, provided that the expression
q'(t) +2p(t)q(t)
2(q(t))³/2
(37)
is a constant. Thus equation (32) can be transformed into
an equation with constant coefficients by a change of the
independent variable, provided that the function (q'+2pq)/9³/2
is a constant.
e. How must the analysis and results in d be modified if
q(t) < 0?
In each of Problems 33 through 36, try to transform the given equation
into one with constant coefficients by the method of Problem 32. If this
is possible, find the general solution of the given equation.
33. y" +ty'+e-1² y = 0,
188
34. y" + 3ty' +1²y = 0,
-∞ < t < ∞
35. ty" + (t²-1)y' + ³y = 0, 0< t << ∞
36. y" +ty' - e-t²y = 0
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