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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each
Suppose that a projectile is fired straight upward with initial velocity v0 from the surface of the earth. If air resistance is not a factor, then its height x(t) at time t satisfies the initial
Suppose that m1 = 2, m2 = 1/2, k1 = 75, k2 = 25, F0 = 100, and ω = 10 (all in mks units) in the forced mass-and- spring system of Fig. 7.5.9. Find the solution of the system Mx" = Kx + F that
In Problems 1 through 16, apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each
Problems 16 through 18 deal with the batted baseball of Example 4, having initial velocity 160 ft/s and air resistance coefficient c = 0.0025.Find the range—the horizontal distance the ball travels
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
In Problems 17 through 25, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of each system. | = 4x1 + x2 +
Problems 16 through 18 deal with the batted baseball of Example 4, having initial velocity 160 ft/s and air resistance coefficient c = 0.0025.Find (to the nearest degree) the initial inclination that
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
In Problems 17 through 25, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of each system. x1 = x1 + 2x2 +
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
Problems 16 through 18 deal with the batted baseball of Example 4, having initial velocity 160 ft/s and air resistance coefficient c = 0.0025.Find (to the nearest half degree) the initial inclination
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each
In Problems 17 through 25, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of each system. x₁ = 4x₁ + x2
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
Find the initial velocity of a baseball hit by Babe Ruth (with c = 0.0025 and initial inclination 40°) if it hit the bleachers at a point 50 ft high and 500 horizontal feet from home plate.
In Problems 17 through 25, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of each system. x3 = 5x1 + x2 +
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
Problems 20 through 23 deal with the same system of three railway cars (same masses) and two buffer springs (same spring constants) as shown in Fig. 7.5.6 and discussed in Example 2. The cars engage
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
Consider the crossbow bolt of Problem 14, fired with the same initial velocity of 288 ft/s and with the air resistance deceleration (0.0002)v2 directed opposite its direction of motion. Suppose that
In Problems 17 through 25, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of each system. x₁ = 5x1-6x3,
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
Suppose that an artillery projectile is fired from ground level with initial velocity 3000 ft/s and initial inclination angle 40°. Assume that its air resistance deceleration is (0.0001)v2.(a) What
Problems 20 through 23 deal with the same system of three railway cars (same masses) and two buffer springs (same spring constants) as shown in Fig. 7.5.6 and discussed in Example 2. The cars engage
(a) Calculate [x (t)]2 + [y(t)]2 to show that the trajectories of the system x' = y, y' = -x of Problem 11 are circles. (b) Calculate [x (t)]2 - [y(t)]2 to show that the trajectories of
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
In Problems 17 through 25, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of each system. - x₁ = 3x₁ +
In Problems 13 through 22, first verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general
(a) Beginning with the general solution of the system x' = -2y, y' = 2x of Problem 13, calculate x2 + y2 to show that the trajectories are circles.(b) Show similarly that the trajectories of the
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
Use the method of Examples 7 and 8 to find the Jordan normal form J of each coefficient matrix A given in Problems 23 through 32 (respectively). Example 7 In Example 3 we saw that the
First solve Eqs. (16) and (17) for e-t and e2t in terms of x(t), y(t), and the constants A and B. Then substitute the results in (e2t) (e-t)2 = 1 to show that the trajectories of the system x' = y,
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.m1 = 1 , m2 = 2; k1 = 2 , k2 = k3 = 4
Apply the method of undetermined coefficients to find a particular solution of each of the systems in Problems 1 through 14. If initial conditions are given, find the particular solution that
Compute the matrix exponential eAt for each system x' = Ax given in Problems 9 through 20.x'1 = 11x1 - 15x2, x'2 = 6x1 - 8x2
Problems 20 through 23 deal with the same system of three railway cars (same masses) and two buffer springs (same spring constants) as shown in Fig. 7.5.6 and discussed in Example 2. The cars engage
In Problems 27 through 29, the system of Fig. 7.5.14 is taken as a model for an undamped car with the given parameters in fps units. (a) Find the two natural frequencies of oscillation (in hertz).
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 22: x1 (0) = 1, x2 (0) = 3, x3 (0) = 4, x4(0) = 7
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
In Problems 17 through 25, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of each system. x₁ = 3x₁ +
In Problems 23 through 32 the eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. Especially in Problems 29 through 32, use of a computer
In Problems 23 through 32, find a particular solution of the indicated linear system that satisfies the given initial conditions.The system of Problem 14: x1 (0) = 0, x2 (0) = 5
Problems 20 through 23 deal with the same system of three railway cars (same masses) and two buffer springs (same spring constants) as shown in Fig. 7.5.6 and discussed in Example 2. The cars engage
The phase portraits in Problems 17 through 28 correspond to linear systems of the form x' = Ax in which the matrix A has two linearly independent eigenvectors. Determine the nature of the eigenvalues
In each of Problems 41 through 46, use the spectral decomposition methods of this section to find a fundamental matrix solution x(t) = eAt for the linear system x' = Ax given in the problem.Problem
Use projection matrices to find a fundamental matrix solution of each of the linear systems given in Problems 1 through 10.The mass-and-spring system of Problem 2, with F1(t) = 96 cos 5t , F2(t) Ξ
Problems 1 and 2 deal with the predator–prey systemthat corresponds to Fig. 9.3.1.Starting with the Jacobian matrix of the system in (1), derive its linearizations at the two critical points (0 ,
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Problems 1 and 2 deal with the predator–prey systemthat corresponds to Fig. 9.3.1.Separate the variables in the quotientof the two equations in (1), and thereby derive the exact implicit solutionof
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx dt = 2x -
In Problems 1 through 4, show that the given system is almost linear with (0 , 0) as a critical point, and classify this critical point as to type and stability. Use a computer system or graphing
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx dt = x -
In Problems 1 through 4, show that the given system is almost linear with (0, 0) as a critical point, and classify this critical point as to type and stability. Use a computer system or graphing
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Let x(t) be a harmful insect population (aphids?) that under natural conditions is held somewhat in check by a benign predator insect population y(t) (ladybugs?). Assume that x(t) and y(t) satisfy
In Problems 1 through 4, show that the given system is almost linear with (0 , 0) as a critical point, and classify this critical point as to type and stability. Use a computer system or graphing
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx dt = x - 2y +
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
In Problems 1 through 4, show that the given system is almost linear with (0 , 0) as a critical point, and classify this critical point as to type and stability. Use a computer system or graphing
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx dt = 2x - 2y -
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Find and classify each of the critical points of the almost linear systems in Problems 5 through 8. Use a computer system or graphing calculator to construct a phase plane portrait that illustrates
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx dt =
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Find and classify each of the critical points of the almost linear systems in Problems 5 through 8. Use a computer system or graphing calculator to construct a phase plane portrait that illustrates
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx dt = 2-4x -
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Find and classify each of the critical points of the almost linear systems in Problems 5 through 8. Use a computer system or graphing calculator to construct a phase plane portrait that illustrates
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx R dt = x -
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Problems 8 through 10 deal with the competition systemin which c1c2 = 8 < 9 = b1b2, so the effect of inhibition should exceed that of competition. The linearization of the system in (3) at (0,0)
Find and classify each of the critical points of the almost linear systems in Problems 5 through 8. Use a computer system or graphing calculator to construct a phase plane portrait that illustrates
In Problems 1 through 8, find the critical point or points of the given autonomous system, and thereby match each system with its phase portrait among Figs. 9.1.12 through 9.1.19. dx dt = x - y - x²
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0 , 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Problems 8 through 10 deal with the competition systemin which c1c2 = 8 < 9 = b1b2, so the effect of inhibition should exceed that of competition. The linearization of the system in (3) at (0,0)
In Problems 1 through 10, apply Theorem 1 to determine the type of the critical point (0, 0) and whether it is asymptotically stable, stable, or unstable. Verify your conclusion by using a computer
Problems 8 through 10 deal with the competition systemin which c1c2 = 8 < 9 = b1b2, so the effect of inhibition should exceed that of competition. The linearization of the system in (3) at (0,0)
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 odd integer, then the critical point (nπ, 0) is a saddle point for the damped pendulum
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 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 nodal sink for the

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