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mathematics
a first course in differential equations

A First Course in Differential Equations with Modeling Applications 10th edition Dennis G. Zill - Solutions

In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{t2 sinh t}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
Find either F(s) or f (t), as indicated.ℒ{t(et + e2t)2}
Use the Laplace transform to solve the given system of differential equations. dy - 2x dt %3D dt dx dy 3x – 3y = 2 dt dt x(0) %3D0, у(0) % 0 +
Use the Laplace transform to solve the given initial-value problem.y" + y = δ(t – 1/2) + δ(t – 3/2 ), y(0) = 0, y'(0) = 0
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. (2t + 1, 0
Find either F(s) or f (t), as indicated.ℒ{t10e-7t}
The function f(t) = (et)10 is not of exponential order. _______Fill in the blanks or answer true or false.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 (a) 1 = L-1 n! (b) " = L- n = 1, 2, 3, ... (c) eat = L-1 %3D k (d) sin kt = L-1 (e) cos kt = L1, s2 + k2 s? + k? k (f) sinh kt = L-1. (g) cosh kt = L-1.
Use the Laplace transform to solve the given system of differential equations. dx + 3x + dt dy = 1 dt dx dy x + -y = e' dt dt x(0) = 0, y(0) = 0
Use the Laplace transform to solve the given initial-value problem.y" + 16y = δ(t – 2π), y(0) = 0,1 (0) = 0
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. t, 0st
In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{ t sin 3t}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{t cos 2t}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
Find either F(s) or f (t), as indicated.ℒ{t3e-2t}
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 48 1.
Use the Laplace transform to solve the given system of differential equations. dx = x - 2y dt dy 5x dt x(0) = -1, y(0) = 2
Use the Laplace transform to solve the given initial-value problem.y'' + y = δ(t – 2π), y(0) = 0, y'(0) = 1
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. (4, 0
In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{t3et}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
Find either F(s) or f (t), as indicated.ℒ{te-6t}
Use the definition of the Laplace transform to find ℒ{f(t)}. (0, 0
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 -1
Use the Laplace transform to solve the given system of differential equations. dx 2y + e dt dy = 8x - t dt x(0) = 1, y(0) = 1
Use the Laplace transform to solve the given initial-value problem.y' + y = δ(t - 1), y(0) = 2
Use the Laplace transform to solve the given initial-value problem.y' - 3y = δ(t - 2), y(0) = 0
Use Definition 7.1.1 to find {f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. -1, 0sI< 1 0st
In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{te-10t}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
Find either F(s) or f(t), as indicated.ℒ{te10t}
Use the definition of the Laplace transform to find ℒ{f(t)}. 0
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1
Use the Laplace transform to solve the given system of differential equations. dx = -x + y dt dy 2x dt x(0) = 0, y(0) = 1
If n is an integer, use the substitution R(x) = (αx)-1/2Z(x) to show that the general solution of the differential equationx2R'' + 2xR'' + [α2x2 - n(n + 1)]R = 0on the interval (0, ∞) is R(x) = c1jn(αx) + c2yn(αx), where jn(αx) and yn(αx) are the spherical Bessel functions of the first and
The differential equation(1 x2)y'' – xy' + a2y = 0where α is a parameter, is known as Chebyshev’s equation after the Russian mathematician Pafnuty Chebyshev (1821–1894). When α = n is a nonnegative integer, Chebyshev’s differential equation always possesses a polynomial solution of
(a) When α = n is a nonnegative integer, Hermite’s differential equation always possesses a polynomial solution of degree n. Use y1(x), given in Problem 51, to find polynomial solutions for n = 0, n = 2, and n = 4.Then use y2(x) to find polynomial solutions for n = 1, n = 3, and n = 5.(b) A
The differential equationy'' - 2xy' + 2αy = 0is known as Hermite’s equation of order α after the French mathematician Charles Hermite (1822-1901). Show that the general solution of the equation is y(x) = c0y1(x), + c1y2(x), whereare power series solutions centered at the ordinary point 0. 24a(a
Use a root-finding application to find the zeros of P1(x), P2(x), . . . , P7 (x). If the Legendre polynomials are built-in functions of your CAS, find zeros of Legendre polynomials of higher degree. Form a conjecture about the location of the zeros of any Legendre polynomial Pn(x), and then
For purposes of this problem ignore the list of Legendre polynomials given on page 266 and the graphs given in Figure 6.4.3. Use Rodrigues’ formula (33) to generate the Legendre polynomials P1(x), P2(x), . . . , P7(x). Use a CAS to carry out the differentiations and simplifications.Figure 6.4.3.
Find the first three positive values of λ for which the problem
Show that the differential equationcan be transformed into Legendre’s equation by means of the substitution x = cos θ. d²y + cos 0 dy + n(n + 1)(sin 0)y = 0 de sin 0 de2
Use the recurrence relation (32) and P0(x) = 1, P1(x) = x, to generate the next six Legendre polynomials.
For the simple pendulum described on page 220 of Section 5.3, suppose that the rod holding the mass m at one end is replaced by a flexible wire or string and that the wire is strung over a pulley at the point of support O in Figure 5.3.3. In this manner, while it is in motion in a vertical plane,
In Example 4 of Section 5.2 we saw that when a constant vertical compressive force, or load, P was applied to a thin column of uniform cross section and hinged at both ends, the deflection y(x) is a solution of the BVP:(a) If the bending stiffness factor EI is proportional to x, then EI(x) = kx,
A uniform thin column of length L, positioned vertically with one end embedded in the ground, will deflect, or bend away, from the vertical under the influence of its own weight when its length or height exceeds a certain critical value. It can be shown that the angular deflection θ(x) of the
(a) Use the general solution obtained in Problem 35 to solve the IVP4'' + tx = 0, x(0.1) = 1, x'(0.1) = 1/2.Use a CAS to evaluate coefficients.(b) Use a CAS to graph the solution obtained in part (a) for 0 ≤ t ≤ 200.
In Problem 21, what do you think is the interval of convergence for the Maclaurin series of sec x?Problem 21In problem the given function is analytic at a = 0. Use appropriate series in (2) and long division to find the first four nonzero terms of the Maclaurin series of the given function.sec x
(a) Use the general solution given in Example 4 to solve the IVP
In Problem 19, find an easier way than multiplying two power series to obtain the Maclaurin series representation of sin x cos x.Problem 19In problem the given function is analytic at a = 0. Use appropriate series in (2) and multiplication to find the first four nonzero terms of the Maclaurin
Use a CAS to graph J3/2(x), J-3/2(x), J5/2(x), and J-5/2(x).
In problem proceed as in Example 4 and find a power series solution
(a) Use (18) to show that the general solution of the differential equation xy'' + λy = 0 on the interval (0, ∞) is(b) Verify by direct substitution that y = √xJ1(2√x) is a particular solution of the DE in the case λ = 1. y = c, VAJ,(2VAx) + c,VãY,(2VAX).
We have seen that x = 0 is a regular singular point of any Cauchy-Euler equation ax2y'' + bxy' + cy = 0. Are the indicial equation (14) for a Cauchy-Euler equation and its auxiliary equation related? Discuss.
In problem proceed as in Example 4 and find a power series solution
Use the Table 6.4.1 to find the first three positive eigenvalues and corresponding eigenfunctions of the boundary-value problem
Each of the differential equationsx3y'' + y = 0 and x2y'' + (3x 1)y' + y = 0has an irregular singular point at x = 0. Determine whether the method of Frobenius yields a series solution of each differential equation about x = 0. Discuss and explain your findings.
In problem proceed as in Example 4 and find a power series solution
(a) Use the result of Problem 34 to express the general solution of Airy’s differential equation for x > 0 in terms of Bessel functions.(b) Verify the results in part (a) using (18).Problem 34Show that y = x1/2 w(2/3 αx3/2) is a solution of Airy’s differential equation y'' + α2xy = 0, x
Discuss how you would define a regular singular point for the linear third-order differential equationa3(x)y''' + a2(x)y'' + a1(x)y' + a0(x)y = 0.
In problem proceed as in Example 4 and find a power series solution
Show that y = x1/2 w(2/3 αx3/2) is a solution of Airy’s differential equation y'' + α2xy = 0, x > 0, whenever w is a solution of Bessel’s equation of order 1/3, that is, t2w'' + tw' + (t2 – 1/9)w = 0, t > 0. After differentiating, substituting, and simplifying, then let t = 2/3 αx3/2.
In Example 4 of Section 5.2 we saw that when a constant vertical compressive force or load P was applied to a thin column of uniform cross section, the deflection y(x) was a solution of the boundary-value problemThe assumption here is that the column is hinged at both ends. The column will buckle
Verify by direct substitution that the given power series is a solution of the indicated differential equation. For a power x2n+1 let k = n + 1. y = = n=02"(n!)2 (-1)" x²", xy" + y' + xy = 0 %3D %3D
Use the change of variablesto show that the differential equation of the aging spring mx'' + ke-at x = 0, α = > 0, becomes 2 -at/2 V m
(a) The differential equation x4y'' + λy = 0 has an irregular singular point at x = 0. Show that the substitution t = 1 x yields the DEwhich now has a regular singular point at t = 0.(b) Use the method of this section to find two series solutions of the second equation in part (a) about the
Verify by direct substitution that the given power series is a solution of the indicated differential equation. For a power x2n+1 let k = n + 1. y = x", (x + 1)y" + y' = 0 n n=1
Use the recurrence relation in Problem 28 along with (23) and (24) to express J3/2(x), J-3/2(x), J5/2(x) and J-5/2(x) in terms of sin x, cos x, and powers of x.Problem 28Use the formula obtained in Example 5 along with part (a) of Problem 27 to derive the recurrence relation2vJv (x) = xJv+1(x) + x
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the recurrence relation found by the method of Frobenius first with the larger root r1. How many solutions did you find? Next use the
Verify by direct substitution that the given power series is a solution of the indicated differential equation. For a power x2n+1 let k = n + 1.
Proceed as on page 264 to derive the elementary form of J-1/2(x) given in (24).
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the recurrence relation found by the method of Frobenius first with the larger root r1. How many solutions did you find? Next use the
Verify by direct substitution that the given power series is a solution of the indicated differential equation. For a power x2n+1 let k = n + 1. (-1)" x2", y' + 2xy = 0 n! y = n=0
In problem use (20) or (21) to obtain the given result.J10(x) = J-1(x) = -J1(x)
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use (23) where necessary and a CAS, if instructed, to find a second
(a) Find one more nonzero term for each of the solutions y1(x) and y2(x) in Example 8.(b) Find a series solution y(x) of the initial-value problem y'' + (cos x)y = 0, y(0) = 1, y'(0) = 1.(c) Use a CAS to graph the partial sums SN (x) for the solution y(x) in part (b). Use N = 2, 3, 4, 5, 6, 7.(d)
In problem proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk.
In problem use (20) or (21) to obtain the given result. rJo(r)dr = xJ (x)
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use (23) where necessary and a CAS, if instructed, to find a second
(a) Find two power series solutions for y'' + xy' + y = 0 and express the solutions y1(x) and y2(x) in terms of summation notation.(b) Use a CAS to graph the partial sums SN (x) for y1(x). Use N = 2, 3, 5, 6, 8, 10. Repeat using the partial sums SN (x) for y2(x).(c) Compare the graphs obtained in
In problem proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk. E n(n – 1)c,x"-2 – 22 nc,x" n=2 n=1 n=0
Use the formula obtained in Example 5 along with part (a) of Problem 27 to derive the recurrence relation2vJv (x) = xJv+1(x) + x Jv-1(x).
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use (23) where necessary and a CAS, if instructed, to find a second
Is x = 0 an ordinary point of the differential equation y'' + 5xy' + √xy = 0?
In problem proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk. E n(n 1)c,x"-2 + E C"+2 n=2 n=0
(a) Proceed as in Example 5 to show thatxJ'υ(x) = -υJυ(x) + xJυ-1(x).Write 2n + υ = 2(n + υ) – υ.(b) Use the result in part (a) to derive (21).
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use (23) where necessary and a CAS, if instructed, to find a second
Is x = 0 an ordinary or a singular point of the differential equation xy" + (sin x)y = 0? Defend your answer with sound mathematics. Use the Maclaurin series of sin x and then examine (sin x)/x.
In problem proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk.
In problem first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of elementary functions.4x2y'' 4xy' + (16x2 + 3)y = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use (23) where necessary and a CAS, if instructed, to find a second
How can the power series method be used to solve the nonhomogeneous equation y" - xy = 1 about the ordinary point x = 0? Of y" - 4xy' - 4y = ex? Carry out your ideas by solving both DEs.
In problem proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk.
(a) Use binomial series to formally show that
In problem first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of elementary functions.16x2y'' + 32xy'' + (x4 - 12)y = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use (23) where necessary and a CAS, if instructed, to find a second
Without actually solving the differential equation (cos x)y" + y' + 5y = 0, find the minimum radius of convergence of power series solutions about the ordinary point x = 0. About the ordinary point x = 1.
(a) From (30) and (31) of Section 6.4 we know that when n = 0, Legendre’s differential equation (1 x2)y'' - 2xy' = 0 has the polynomial solution y = P0(x) = 1. Use (5) of Section 4.2 to show that a second Legendre function satisfying the DE for 1 < x < 1 is(b) We also know from (30) and
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ∞).2x2y'' +
In problem use the procedure in Example 8 to find two power series solutions of the given differential equation about the ordinary point x = 0.y'' + exy' – y = 0
Use a substitution to shift the summation index so that the general term of given power series involves xk.
(a) Use (10) of Section 6.4 and Problem 32 of Exercises 6.4 to show that(b) Use (15) of Section 6.4 to show that(c) Use (16) of Section 6.4 and part (b) to show that 2 (cos: Y3/2(x) + sin x TTX
In problem first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of elementary functions.x2y' + 4xy' + (x2 + 2)y = 0

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