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study help
mathematics
first course differential equations
Questions and Answers of
First Course Differential Equations
Problems 11 through 13 deal with the predator–prey systemin which the prey population x (t) is logistic but the predator population y(t) would (in the absence of any prey) decline naturally.
Each of the systems in Problems 11 through 18 has a single critical point (x0 , y0). Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a
In each of Problems 12 through 16, a second-order equation of the form x'' + f (x, x') = 0, corresponding to a certain mass-and-spring system, is given. Find and classify the critical points of the
In Problems 9 through 12, find each equilibrium solution x(t) Ξ x0 of the given second-order differential equation x" + f(x , x') = 0. Use a computer system or graphing calculator to construct a
Problems 9 through 11 deal with the damped pendulum system x' = y, y' = -ω2 sinx - cy.Show that if n is an even integer and c2 > 4ω2, then the critical point (nπ, 0) is a spiral sink for the
Problems 11 through 13 deal with the predator–prey systemin which the prey population x (t) is logistic but the predator population y(t) would (in the absence of any prey) decline naturally.
Each of the systems in Problems 11 through 18 has a single critical point (x0 , y0). Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a
In each of Problems 12 through 16, a second-order equation of the form x'' + f (x, x') = 0, corresponding to a certain mass-and-spring system, is given. Find and classify the critical points of the
Problems 11 through 13 deal with the predator–prey systemin which the prey population x (t) is logistic but the predator population y(t) would (in the absence of any prey) decline naturally.
In Problems 9 through 12, find each equilibrium solution x(t) Ξ x0 of the given second-order differential equation x" + f(x , x') = 0. Use a computer system or graphing calculator to construct a
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Each of the systems in Problems 11 through 18 has a single critical point (x0 , y0). Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a
In each of Problems 12 through 16, a second-order equation of the form x'' + f (x, x') = 0, corresponding to a certain mass-and-spring system, is given. Find and classify the critical points of the
Problems 14 through 17 deal with the predator–prey systemHere each population-the prey population x(t) and the predator population y(t)-is an unsophisticated population (like the alligators of
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Each of the systems in Problems 11 through 18 has a single critical point (x0 , y0). Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a
In each of Problems 12 through 16, a second-order equation of the form x'' + f (x, x') = 0, corresponding to a certain mass-and-spring system, is given. Find and classify the critical points of the
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Each of the systems in Problems 11 through 18 has a single critical point (x0 , y0). Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a
Problems 14 through 17 deal with the predator–prey systemHere each population-the prey population x(t) and the predator population y(t)-is an unsophisticated population (like the alligators of
In each of Problems 12 through 16, a second-order equation of the form x'' + f (x, x') = 0, corresponding to a certain mass-and-spring system, is given. Find and classify the critical points of the
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Problems 14 through 17 deal with the predator–prey systemHere each population-the prey population x(t) and the predator population y(t)-is an unsophisticated population (like the alligators of
Each of the systems in Problems 11 through 18 has a single critical point (x0 , y0). Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a
In Problems 17 through 20, analyze the critical points of the indicated system, use a computer system to construct an illustrative position–velocity phase plane portrait, and describe the
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Problems 14 through 17 deal with the predator–prey systemHere each population-the prey population x(t) and the predator population y(t)-is an unsophisticated population (like the alligators of
Each of the systems in Problems 11 through 18 has a single critical point (x0 , y0). Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a
In Problems 11 through 14, use the method of Example 4 to find two linearly independent power series solutions of the given differential equation. Determine the radius of convergence of each series,
Use the method of Example 6 to find two linearly independent Frobenius series solutions of the differential equations in Problems 27 through 31. Then construct a graph showing their graphs for x >
In Problems 23 through 26, find a three-term recurrence relation for solutions of the form y = Σ cnxn. Then find the first three nonzero terms in each of two linearly independent solutions.(1 + x3)
Solve the initial value problemDetermine sufficiently many terms to compute y(1/2) accurate to four decimal places. y" + xy' + (2x² + 1)y=0; y(0) = 1, y'(0) = -1.
Use the method of Example 6 to find two linearly independent Frobenius series solutions of the differential equations in Problems 27 through 31. Then construct a graph showing their graphs for x >
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.16x2y" - (5 - 144x3) y = 0
Use the method of Example 6 to find two linearly independent Frobenius series solutions of the differential equations in Problems 27 through 31. Then construct a graph showing their graphs for x >
(a) Show that the substitutiontransforms the Riccati equation dy/dx = x2 + y2 into u" + x2u = 0.(b) Show that the general solution of dy/dx = x2 + y2 isApply the identities in Eqs. (22) and (23). 1
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.2x2y" - 3xy' - 2(14 - x5)y = 0
Apply Theorem 1 to show that the general solution ofis y(x) = x-1(A cos x + B sin x). xy" + 2y + xy = 0
In Problems 28 through 30, find the first three nonzero terms in each of two linearly independent solutions of the form y = Σ cnxn. Substitute known Taylor series for the analytic functions and
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.y'' + x4 y = 0
Use the method of Example 6 to find two linearly independent Frobenius series solutions of the differential equations in Problems 27 through 31. Then construct a graph showing their graphs for x >
In Problems 28 through 30, find the first three nonzero terms in each of two linearly independent solutions of the form y = Σ cnxn. Substitute known Taylor series for the analytic functions and
Use the method of Example 6 to find two linearly independent Frobenius series solutions of the differential equations in Problems 27 through 31. Then construct a graph showing their graphs for x >
In Problems 28 through 30, find the first three nonzero terms in each of two linearly independent solutions of the form y = Σ cnxn. Substitute known Taylor series for the analytic functions and
In Problems 19 through 30, express the general solution of the given differential equation in terms of Bessel functions.y" + 4x3 y = 0
Verify that the substitutions in (2) in Bessel’s equation [Eq. (1)] yield Eq. (3). x²y" + xy + (x² - p²) y = 0. (1)
In Problems 32 through 34, find the first three nonzero terms of each of two linearly independent Frobenius series solutions.2x2y" + x(x + 1)y' - (2x + 1)y = 0
Apply the method of Frobenius to Bessel’s equation of order 1/2,to derive its general solution for x > 0,Figure 11.3.2 shows the graphs of the two indicated solutions. x²y" + xy + (x² - 1) y =
Note that x = 0 is an irregular point of the equation(a) Show that y = xr Σ cnxn can satisfy this equation only if r = 0.(b) Substitute y = Σ cnxn to derive the "formal" solution y = Σ n!xn. What
In Problems 32 through 34, find the first three nonzero terms of each of two linearly independent Frobenius series solutions.(2x2 + 5x3)y" + (3x - x2)y' - (1 + x) y = 0
Problems 18 through 25 deal with the predator–prey systemfor which a bifurcation occurs at the value ∈ = 0 of the parameter ∈ . Problems 18 and 19 deal with the case ∈ = 0, in which case the
(a) Suppose that A and B are nonzero constants. Show that the equation x2y" + Ay' + By = 0 has at most one solution of the form y = xr Σcnxn.(b) Repeat part (a) with the equation x3y" + Axy' + By =
In Problems 32 through 34, find the first three nonzero terms of each of two linearly independent Frobenius series solutions.2x2y'' + (sin x)y' - (cos x)y = 0
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Problems 18 through 25 deal with the predator–prey systemfor which a bifurcation occurs at the value ∈ = 0 of the parameter ∈ . Problems 18 and 19 deal with the case ∈ = 0, in which case the
In Problems 17 through 20, analyze the critical points of the indicated system, use a computer system to construct an illustrative position–velocity phase plane portrait, and describe the
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
In Problems 17 through 20, analyze the critical points of the indicated system, use a computer system to construct an illustrative position–velocity phase plane portrait, and describe the
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
In Problems 23 through 26, a system dx/dt = F(x, y), dy/dt = G(x, y) is given. Solve the equationto find the trajectories of the given system. Use a computer systemor graphing calculator to construct
Problems 23 through 25 deal with the case ∈ = 1, so that the system in (6) takes the formand these problems imply that the three critical points (0,0), (7,0), and (5,2) of the system in (9) are as
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
The term bifurcation generally refers to something “splitting apart.” With regard to differential equations or systems involving a parameter, it refers to abrupt changes in the character of the
In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f (t). f(t)= = 1 - cos 2t t
Apply Theorem 2 to find the inverse Laplace transforms of the functions in Problems 17 through 24. F(s) = 1 s² (s² + 1)
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = t cos 2t t f(t) t" (n
In Problems 15 through 22, apply either Theorem 2 or Theorem 3 to find the Laplace transform of f (t). f(t) = et - e-t t
Use the transforms in Fig. 10.1.2 to find the Laplace transforms of the functions in Problems 11 through 22. A preliminary integration by parts may be necessary. f (t) = sinh2 3t t f(t) t" (n
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Problems 20 through 22 deal with the case ∈ = -1, for which the system in (6) becomesand imply that the three critical points (0,0), (3,0), and (5,2) of (8) are as shown in Fig. 9.3.17-with a nodal
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
Solve each of the linear systems in Problems 13 through 20 to determine whether the critical point (0, 0) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator
Problems 20 through 22 deal with the case ∈ = -1, for which the system in (6) becomesand imply that the three critical points (0,0), (3,0), and (5,2) of (8) are as shown in Fig. 9.3.17-with a nodal
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
Problems 20 through 22 deal with the case ∈ = -1, for which the system in (6) becomesand imply that the three critical points (0,0), (3,0), and (5,2) of (8) are as shown in Fig. 9.3.17-with a nodal
Separate variables in Eq. (20) to derive the solution in (21). dr dt =r(1-²). (20)
In Problems 23 through 26, a system dx/dt = F(x, y), dy/dt = G(x, y) is given. Solve the equationto find the trajectories of the given system. Use a computer system or graphing calculator to
In Problems 19 through 28, investigate the type of the critical point (0, 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct a
Problems 23 through 25 deal with the case ∈ = 1, so that the system in (6) takes the formand these problems imply that the three critical points (0,0), (7,0), and (5,2) of the system in (9) are as
In Problems 23 through 26, a system dx/dt = F(x, y), dy/dt = G(x, y) is given. Solve the equationto find the trajectories of the given system. Use a computer systemor graphing calculator to construct
Problems 23 through 25 deal with the case ∈ = 1, so that the system in (6) takes the formand these problems imply that the three critical points (0,0), (7,0), and (5,2) of the system in (9) are as
In Problems 23 through 26, a system dx/dt = F(x, y), dy/dt = G(x, y) is given. Solve the equationto find the trajectories of the given system. Use a computer systemor graphing calculator to construct
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
For each two-population system in Problems 26 through 34, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
For each two-population system in Problems 26 through 34, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
Let (x(t) , y(t)) be a nontrivial solution of the nonautonomous systemSuppose that ϕ(t) = x (t + ϒ) and Ψ(t) = y(t + ϒ), where y ≠ 0. Show that (ϕ(t), Ψ (t)) is not a solution of the system.
For each two-population system in Problems 26 through 34, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
In Problems 19 through 28, investigate the type of the critical point (0 , 0) of the given almost linear system. Verify your conclusion by using a computer system or graphing calculator to construct
For each two-population system in Problems 26 through 34, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
In Problems 29 through 32, find all critical points of the given system, and investigate the type and stability of each. Verify your conclusions by means of a phase portrait constructed using a
For each two-population system in Problems 26 through 34, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
In Problems 29 through 32, find all critical points of the given system, and investigate the type and stability of each. Verify your conclusions by means of a phase portrait constructed using a
For each two-population system in Problems 26 through 34, first describe the type of x-and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
For each two-population system in Problems 26 through 34, first describe the type of x-and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
In Problems 29 through 32, find all critical points of the given system, and investigate the type and stability of each. Verify your conclusions by means of a phase portrait constructed using a
In Problems 29 through 32, find all critical points of the given system, and investigate the type and stability of each. Verify your conclusions by means of a phase portrait constructed using a
For each two-population system in Problems 26 through 34, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction— competition,
In the case of a two-dimensional system that is not almost linear, the trajectories near an isolated critical point can exhibit a considerably more complicated structure than those near the nodes,
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