- 22.214.171.124.36: Verify the calculations in Example I.
- 126.96.36.199.37: Solve by the Euler method:y~ = -3)"1 + )"2' y~ = Y1 - 3)"2' )"1(0) ...
- 188.8.131.52.38: Solve by the Euler method:y~ = )"1, Y; = Y2, Y1(0) = I, )'2(0) = -1...
- 184.108.40.206.39: Solve by the Euler method:Y~ = Yb Y; = -J'2, .VI(O) = 2, Y2(0) = 2,...
- 220.127.116.11.40: Solve by the Euler method:y" + 4y = 0, yeO) = 1, y'(O) = 0, h 0.2,5...
- 18.104.22.168.41: Solve by the Euler method:y" - )'5 stepsx, nO) 1, y' (0) -2, h 0.1,
- 22.214.171.124.42: Solve by the Euler method:y~ = -.\"1 + Y2' Y; = -.\"1 - Y2, ."1(0) ...
- 126.96.36.199.43: Verify the formulas and calculations for the Airy equation in Examp...
- 188.8.131.52.44: Solve by the classical RK: The system in Prob. 7. How much smaller ...
- 184.108.40.206.45: Solve by the classical RK:The ODE in Prob. 6. By what factor did th...
- 220.127.116.11.46: Solve by the classical RK: Undamped Pendulum. Y" + siny = 0, yt77) ...
- 18.104.22.168.47: Solve by the classical RK:Bessel Function 10, xy" + y' + xy = 0,y(l...
- 22.214.171.124.48: Solve by the classical RK: .r ~ = -4Y1 + )'2' y~ = .\'1 - 4)'2' YI(...
- 126.96.36.199.49: Solve by the classical RK:The system in Prob. 2. Ho'W much smaller ...
- 188.8.131.52.50: Verify the calculations for the Airy equation in Example 3.
- 184.108.40.206.51: Do by RKN:Prob. 12 (Bessel function Jo). Compare the results.
- 220.127.116.11.52: Do by RKN:,," - n" + 4,' = 0, ,'(0) = 3, ,,' (0) = 0,i, = 0.2: 5 st...
- 18.104.22.168.53: Do by RKN: (x 2 - x)y" - xy' + y = 0, y(!) = I - ! In 2.y'(!) = I -...
- 22.214.171.124.54: Do by RKN:Prob. II. Compare the results.
- 126.96.36.199.55: CAS EXPERIMENT. Comparison of Methods. (a) Write program~ for RKN a...
- 188.8.131.52.56: CAS EXPERIMENT. Backward Euler and Stiffness. Extend Example 4 as f...
Solutions for Chapter 21.3: Methods for Systems and Higher Order ODEs
Full solutions for Advanced Engineering Mathematics | 9th Edition
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Outer product uv T
= column times row = rank one matrix.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Solvable system Ax = b.
The right side b is in the column space of A.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.