Questions and Answers of A First Course in Differential Equations

In problem use the Adams-Bashforth-Moulton method to approximate y(1.0), where y(x) is the solution of the given initial-value problem. First use h = 0.2 and then use h = 0.1. Use the RK4 method to
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = e-y, y(0) = 0; y(0.5)
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = (x - y)2, y(0) = 0.5; y(0.5)
Use the Adams-Bashforth-Moulton method to approximate y(0.4), where y(x) is the solution of the initialvalue problem y' = 4x - 2y, y(0) = 2. Use h = 0.1 and the RK4 method to compute y1, y2, and y3.
When E = 100 V, R = 10 Ω, and L = 1 h, the system of differential equations for the currents i1(t) and i3(t) in the electrical network given in Figure 9.4.3 iswhere i1(0) = 0 and i3(0) [1] 0. Use
In problem use the Adams-Bashforth-Moulton method to approximate y(1.0), where y(x) is the solution of the given initial-value problem. First use h = 0.2 and then use h = 0.1. Use the RK4 method to
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = x2 + y2, y(0) = 1; y(0.5)
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = x + y2, y(0) = 0; y(0.5)
Use Euler’s method to approximate y(0.2), where y(x) is the solution of the initial-value problemy'' - (2x + 1)y = 1, y(0) = 3, y'(0) = 1. First use one step with h [1] 0.2 and then repeat the
Use the RK4 method to approximate y(0.2), where y(x) is the solution of the initial-value problemy'' - 2y' + 2y = et cos t, y(0) = 1, y'(0) [1] 2.First use h = 0.2 and then use h = 0.1.
In problem use the Adams-Bashforth-Moulton method to approximate y(1.0), where y(x) is the solution of the given initial-value problem. First use h = 0.2 and then use h = 0.1. Use the RK4 method to
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = 1 + y2, y(0) = 0; y(0.5)
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = e-y , y(0) = 0; y(0.5)
In problem 1-4 construct a table comparing the indicated values of y(x) using Euler’s method, the improved Euler’s method, and the RK4 method. Compute to four rounded decimal places. First use h
In problem repeat the indicated problem using the RK4 method. First use h = 0.2 and then use h = 0.1.Problem 2Use Euler’s method to approximate y(1.2), where y(x) is the solution of the
In problem use the Adams-Bashforth-Moulton method to approximate y(0.8), where y(x) is the solution of the given initial-value problem. Use h = 0.2 and the RK4 method to compute y1, y2, and y3.y' =
In problem use the RK4 method with h = 0.1 to obtain a four-decimal approximation of the indicated value.y' = 4x - 2y, y(0) = 2; y(0.5)
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = X2 ± Y2, Y(0) = 1; Y(0.5)
In problems 1-4 construct a table comparing the indicated values of y(x) using Euler’s method, the improved Euler’s method, and the RK4 method. Compute to four rounded decimal places. First use h
In problem repeat the indicated problem using the RK4 method. First use h = 0.2 and then use h = 0.1.Problem 1Use Euler’s method to approximate y(0.2), where y(x) is the solution of the
In problem use the Adams-Bashforth-Moulton method to approximate y(0.8), where y(x) is the solution of the given initial-value problem. Use h = 0.2 and the RK4 method to compute y1, y2, and y3.y' =
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = 1 + y2, y(0) = 0; y(0.5)
In =roblem construct a table comparing the indicated values of y(x) using Euler’s method, the improved Euler’s method, and the RK4 method. Compute to four rounded decimal places. First use h =
Use Euler’s method to approximate y(1.2), where y(x) is the solution of the initial-value problemx2 y'' - 2xy' + 2y = 0, y(1) = 4, y'(1) = 9,where x > 0. Use h = 0.1. Find the analytic solution
Write a computer program to implement the Adams- Bashforth-Moulton method.
Assume w2 = 3/4 that in (4). Use the resulting secondorder Runge-Kutta method to approximate y(0.5), where y(x) is the solution of the initial-value problem in Problem 1. Compare this approximate
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = 4x - 2y, y(0) = 2; y(0.5)
In problem construct a table comparing the indicated values of y(x) using Euler’s method, the improved Euler’s method, and the RK4 method. Compute to four rounded decimal places. First use h =
Use Euler’s method to approximate y(0.2), where y(x) is the solution of the initial-value problemy'' - 4y' + 4y = 0, y (0) = -2, y'(0) = 1.Use h = 0.1. Find the analytic solution of the problem,
Find the analytic solution of the initial-value problem in Example 1. Compare the actual values of y(0.2), y(0.4), y(0.6), and y(0.8) with the approximations y1, y2, y3, and y4.
Use the RK4 method with h = 0.1 to approximate y(0.5), where y(x) is the solution of the initial-value problem y' = (x + y - 1)2, y(0) = 2. Compare this approximate value with the actual value
In problem use the improved Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.y' = 2x - 3y + 1, y(1) = 5; y(1.5)
Examine your phase portraits in Problem 47. Under what conditions will the phase portrait of a 2 x 2 homogeneous linear system with complex eigenvalues consist of a family of closed curves? consist
Obtain a Cartesian equation of the curve defined parametrically by the solution of the linear system in Example 6. Identify the curve passing through (2, -1) in Figure 8.2.5. Compute x2, y2, and
Consider the 5 × 5 matrix given in Problem 31. Solve the system ' = AX without the aid of matrix methods, but write the general solution using matrix notation. Use the general solution as a basis
Solve each of the following linear systems.a.b.Find a phase portrait of each system. What is the geometric significance of the line y = -x in each portrait? X' = 1
(a) Solve (2) of Section 7.6 using the first method outlined in the Remarks (page 345)—that is, express (2) of Section 7.6 as a first-order system of four linear equations. Use a CAS or linear
Find phase portraits for the systems in Problems 36, 37, and 38.Problems 36, 37, and 38.Find the general solution of the given system. X' = -4.
Solve the given initial-value problem. -2 X' = 15 Х. ХО) 4 X(0) 8/
Solve the given initial-value problem. -12 -14) X' = |1 2 -3 X, X(0) = 6. 1 -2) -7
Find the general solution of the given system. 2 4 X' =-1 -2 0 -2/ 0 X -1
Find the general solution of the given system. 5 1' -6 4 X 0 0 2 2 5 X' =-5
Find the general solution of the given system. 1 -1 2) 1 0X X' =-1 -1 0 1
Find the general solution of the given system. dx 2x + y + 2z dt dy 3x + 6z dt dz -4x – 3z dt
Find the general solution of the given system. 一Z y |如山山|山水山
Find the general solution of the given system. X' = (; ) 1 -3,
Find the general solution of the given system. 4 X' = -4.
Find the general solution of the given system. dx 4x + 5y dt dy -2x + 6y dt
Solving a nonhomogeneous linear system X' = AX + F(t) by variation of parameters when A is a 3 × 3 (or larger) matrix is almost an impossible task to do by hand. Consider the system(a) Use a CAS or
Find the general solution of the given system. dx 5x + y dt dy -2x + 3y dt %3D
If y1 and y2 are linearly independent solutions of the associated homogeneous DE for y'' + P(x)y' + Q(x)y = f (x), show in the case of a nonhomogeneous linear second-order DE that (9) reduces to the
Find the general solution of the given system. dx = x + y di dy -2х — у dt
The system of differential equations for the currents i1(t) and i2(t) in the electrical network shown in Figure 8.3.2 isUse variation of parameters to solve the system if R1 = 8 Ω, R2 = 3 Ω, L1 = 1
Find the general solution of the given system. dx = 6x - y dt dy 5x + 2y dt
In problem use (14) to solve the given initialvalue problem. x' = (; )x• G) xo=(-) X' = )x + X(1) = 1
Find phase portraits for the systems in Problems 20 and 21. For each system find any half-line trajectories and include these lines in your phase portrait.Problems 20 and 21. dx -6x + 5y dt -5x + 4y
In problem use (14) to solve the given initialvalue problem. X' = 3 -1 4e2 X + X(0) = 3 4et
Show that the 5 × 5 matrixhas an eigenvalue λ1 of multiplicity 5. Show that three linearly independent eigenvectors corresponding to λ1 can be found. 2 1 00 0' 0 2 0 0 0 A = 0 0 2 0 0 0 0 0 0 2 1
In problem use variation of parameters to solve the given system. 3 -1 -1 X' =1 X + 1 -1 1/ 2e/
Solve the given initial-value problem. /0 0 1 X' = 0 1 0X, X(0) = |2 10 0
In problem use variation of parameters to solve the given system.
Solve the given initial-value problem. 2 4 X, X(0) = -1 6/ X' = 6)
In problem use variation of parameters to solve the given system. X' = (;) tan i X +
Find the general solution of the given system. /4 1 0\ X' = 0 4 4 1X 0 0 4/ 0 4/
Use (1) to find the general solution ofUse MATLAB or a CAS to find eAt. -4 0 6 X' = -1 -5 0 -4 X. 0 1 3 0 2/
In problem use variation of parameters to solve the given system. 1 X + csc c t X' = sec t
Find the general solution of the given system. /1 0 0\ X' = 2 2 -1 X 0 1 0/
In problem use variation of parameters to solve the given system. X' = ( ); X + -1 cot t)
Find the general solution of the given system. X' = 0 3 1X 0 -1 1
Prove that the general solution ofon the interval (-∞, ∞) is X' = -6
(a) Use (1) to find the general solution ofUse a CAS to find eAt. Then use the computer to find eigenvalues and eigenvectors of the coefficient matrixand form the general solution in the manner of
A matrix A is said to be nilpotent if there exists some positive integer m such that Am = 0. Verify thatis nilpotent. Discuss why it is relatively easy to compute eAt when A is nilpotent. Compute eAt
In problem use variation of parameters to solve the given system. X' = (: X + 0, -1 sec t tan t,
Find the general solution of the given system. (5 -4 0V X' = 1 0 2 X 2 5
Prove that the general solution ofon the interval (-∞, ∞) is (0 6 0 X' = 1 0 1X 11 0
Reread the discussion leading to the result given in (7). Does the matrix sI - A always have an inverse? Discuss.
In problem use variation of parameters to solve the given system. X' = (2 -2 X + %3D 8 -6, t
Find the general solution of the given system. dx 3x + 2y + 4z dt dy 2x + 2z dt dz 4x + 2y + 3z dt
Verify that the vector Xp is a particular solution of the given system. 1 2 3 X' =|-4 2 0X + -6 1 0/ sin 3t 4 sin 3t; X, = 3 cos 3t
Solve the given system. X' = %3D 1 2.
Find the general solution of the given system. dx = 3x – y - z dt dy = x + y - z dt dz = x - y + z dt
Verify that the vector Xp is a particular solution of the given system. X' = 3 x-G x- () x.-()- -() |e'; X, =
Solve the given system. 2 1 X' = -3 6
In problem use variation of parameters to solve the given system. dx = 3x – 3y + 4 dt dy = 2x – 2y - 1 dt
In problem solve the given linear system. X' X + e tan tan t)
Verify that the vector X is a solution of the given system. dx -2x + 5y dt %3D 5 cos t dy -2r + 4y; X = dt 3 cos t - sin t
Find the general solution of the given system. 2 -1 -2 X 6| -1 4 X' =
In problem use variation of parameters to solve the given system. dx = 2x - y dt dy = 3x – 2y + 4t dt
Solve the system in Problem 7 subject to the initial conditionProblem 7In problem use (1) to find the general solution of the given system. X(0) -4 6/
In problem solve the given linear system. (-1 X' = -2 )x + (o) (cot t/
Verify that the vector X is a solution of the given system. x -( x-( X' = - 3t/2 -1,
Solve the given initial-value problem. X' 3. X, X(0) = =
In problem use variation of parameters to solve the given system. (3 -5 X' = X +
Solve the system in Problem 9 subject to the initial conditionProblem 9In problem use (5) to find the general solution of the given system. X(0) 3)
In problem solve the given linear system. 3 X' = -1 -2' X +
Verify that the vector X is a solution of the given system. X' = ( 2 X; X = et te 4
Solve the given initial-value problem. (1 1 4 X' = 0 2 0 X, X(0) = 3 1 1 1
In problem use variation of parameters to solve the given system. x' = (; )x+ () cos t et sin t
In problem use variation of parameters to solve the given system. X' = (; )* + (2cos 2) sin 21 ,21 2 cos 2t,
In problem use the method of Example 2 to compute eAt for the coefficient matrix. Use (1) to find the general solution of the given system. X' = 4 3 -4 -4,

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