19. 12y" +2ty' - 2y = 0, t> 0; y₁(t) = t 20 2," +201²+ ( 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
19
![16. Suppose that r₁ and r2 are roots of ar2 + br + c = 0 and that
r₁ r2; then exp(r₁t) and exp(r2t) are solutions of the differential
equation ay" +by' + cy= 0. Show that
e'2¹ - e'1¹
12 - 1
is also a solution of the equation for r2 r₁. Then think of r, as fixed,
and use l'Hôpital's rule to evaluate the limit of (1; r1, 2) as r2 → r₁,
thereby obtaining the second solution in the case of equal roots.
17. a. If ar2 + br + c = 0 has equal roots r₁, show that
L[e"]= a(e")" + b(e")' +ce" = a(r-r₁) ²e¹¹. (37)
(t; r1, 12): =
Since the right-hand side of equation (37) is zero when r = r₁, it
follows that exp(r₁t) is a solution of L[y] =ay" +by' +cy = 0.
b. Differentiate equation (37) with respect to r, and interchange
differentiation with respect to r and with respect to t, thus
showing that
0, 1L [e" ] = L [ en ] = L [te"]
Ər
= ate" (r = r₁)² + 2ae"' (r = r₁).
(38)
Since the right-hand side of equation (36) is zero when r = r₁,
conclude that t exp(r₁t) is also a solution of L[y] = 0.
In each of Problems 18 through 22, use the method of reduction of
order to find a second solution of the given differential equation.
18. 12y" - 4ty' + 6y = 0, t> 0; y₁(t) = t²
19. t2y" +2ty' - 2y = 0, t> 0;
20. 12y" + 3ty' + y = 0, t> 0;
21. xy" - y' +4x³y = 0, x > 0;
22. x²y"+xy'+(x²-0.25) y = 0,
23. The differential equation
y₁(t) = t
y₁(t) = t-1
y₁(x) = sin(x²)
x > 0; y₁(x) = x-1/2 sin x
y" + 6(xy' + y) = 0
=
arises in the study of the turbulent flow of a uniform stream past a
circular cylinder. Verify that y₁(x) exp(-6x2/2) is one solution,
and then find the general solution in the form of an integral.
24. The method of Problem 15 can be extended to second-order
equations with variable coefficients. If y₁ is a known nonvanishing
solution of y" + p(t) y' +q(t) y = 0, show that 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.
y₁(t) = t-1
y₁ (t) = sin(1²)
25. 12y" + 3ty' + y = 0, t> 0;
26. ty" - y' +4t³y = 0, t> 0;
y₁(x) = x-1/2 sin x
27. x2y"+xy'+(x2-0.25) y = 0, x > 0;
Behavior of Solutions as →∞o. Problems 28 through 30 are
concerned with the behavior of solutions as t → ∞o.
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
30.
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→∞o. Determine this constant for the initial conditions
y(0) = yo, y'(0) = yo.
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 sint ≥ 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→∞o. 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. t2y" - 3ty' + 4y = 0, t> 0
32. 12y" +2ty' +0.25y = 0, t> 0
33. 12y" + 3ty' + y = 0, t> 0
34. 4t2y" - 8ty' +9y = 0, t>0
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Transcribed Image Text:16. Suppose that r₁ and r2 are roots of ar2 + br + c = 0 and that
r₁ r2; then exp(r₁t) and exp(r2t) are solutions of the differential
equation ay" +by' + cy= 0. Show that
e'2¹ - e'1¹
12 - 1
is also a solution of the equation for r2 r₁. Then think of r, as fixed,
and use l'Hôpital's rule to evaluate the limit of (1; r1, 2) as r2 → r₁,
thereby obtaining the second solution in the case of equal roots.
17. a. If ar2 + br + c = 0 has equal roots r₁, show that
L[e"]= a(e")" + b(e")' +ce" = a(r-r₁) ²e¹¹. (37)
(t; r1, 12): =
Since the right-hand side of equation (37) is zero when r = r₁, it
follows that exp(r₁t) is a solution of L[y] =ay" +by' +cy = 0.
b. Differentiate equation (37) with respect to r, and interchange
differentiation with respect to r and with respect to t, thus
showing that
0, 1L [e" ] = L [ en ] = L [te"]
Ər
= ate" (r = r₁)² + 2ae"' (r = r₁).
(38)
Since the right-hand side of equation (36) is zero when r = r₁,
conclude that t exp(r₁t) is also a solution of L[y] = 0.
In each of Problems 18 through 22, use the method of reduction of
order to find a second solution of the given differential equation.
18. 12y" - 4ty' + 6y = 0, t> 0; y₁(t) = t²
19. t2y" +2ty' - 2y = 0, t> 0;
20. 12y" + 3ty' + y = 0, t> 0;
21. xy" - y' +4x³y = 0, x > 0;
22. x²y"+xy'+(x²-0.25) y = 0,
23. The differential equation
y₁(t) = t
y₁(t) = t-1
y₁(x) = sin(x²)
x > 0; y₁(x) = x-1/2 sin x
y" + 6(xy' + y) = 0
=
arises in the study of the turbulent flow of a uniform stream past a
circular cylinder. Verify that y₁(x) exp(-6x2/2) is one solution,
and then find the general solution in the form of an integral.
24. The method of Problem 15 can be extended to second-order
equations with variable coefficients. If y₁ is a known nonvanishing
solution of y" + p(t) y' +q(t) y = 0, show that 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.
y₁(t) = t-1
y₁ (t) = sin(1²)
25. 12y" + 3ty' + y = 0, t> 0;
26. ty" - y' +4t³y = 0, t> 0;
y₁(x) = x-1/2 sin x
27. x2y"+xy'+(x2-0.25) y = 0, x > 0;
Behavior of Solutions as →∞o. Problems 28 through 30 are
concerned with the behavior of solutions as t → ∞o.
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
30.
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→∞o. Determine this constant for the initial conditions
y(0) = yo, y'(0) = yo.
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 sint ≥ 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→∞o. 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. t2y" - 3ty' + 4y = 0, t> 0
32. 12y" +2ty' +0.25y = 0, t> 0
33. 12y" + 3ty' + y = 0, t> 0
34. 4t2y" - 8ty' +9y = 0, t>0
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