 21.3.21.1.36: Verify the calculations in Example I.
 21.3.21.1.37: Solve by the Euler method:y~ = 3)"1 + )"2' y~ = Y1  3)"2' )"1(0) ...
 21.3.21.1.38: Solve by the Euler method:y~ = )"1, Y; = Y2, Y1(0) = I, )'2(0) = 1...
 21.3.21.1.39: Solve by the Euler method:Y~ = Yb Y; = J'2, .VI(O) = 2, Y2(0) = 2,...
 21.3.21.1.40: Solve by the Euler method:y" + 4y = 0, yeO) = 1, y'(O) = 0, h 0.2,5...
 21.3.21.1.41: Solve by the Euler method:y"  )'5 stepsx, nO) 1, y' (0) 2, h 0.1,
 21.3.21.1.42: Solve by the Euler method:y~ = .\"1 + Y2' Y; = .\"1  Y2, ."1(0) ...
 21.3.21.1.43: Verify the formulas and calculations for the Airy equation in Examp...
 21.3.21.1.44: Solve by the classical RK: The system in Prob. 7. How much smaller ...
 21.3.21.1.45: Solve by the classical RK:The ODE in Prob. 6. By what factor did th...
 21.3.21.1.46: Solve by the classical RK: Undamped Pendulum. Y" + siny = 0, yt77) ...
 21.3.21.1.47: Solve by the classical RK:Bessel Function 10, xy" + y' + xy = 0,y(l...
 21.3.21.1.48: Solve by the classical RK: .r ~ = 4Y1 + )'2' y~ = .\'1  4)'2' YI(...
 21.3.21.1.49: Solve by the classical RK:The system in Prob. 2. Ho'W much smaller ...
 21.3.21.1.50: Verify the calculations for the Airy equation in Example 3.
 21.3.21.1.51: Do by RKN:Prob. 12 (Bessel function Jo). Compare the results.
 21.3.21.1.52: Do by RKN:,,"  n" + 4,' = 0, ,'(0) = 3, ,,' (0) = 0,i, = 0.2: 5 st...
 21.3.21.1.53: Do by RKN: (x 2  x)y"  xy' + y = 0, y(!) = I  ! In 2.y'(!) = I ...
 21.3.21.1.54: Do by RKN:Prob. II. Compare the results.
 21.3.21.1.55: CAS EXPERIMENT. Comparison of Methods. (a) Write program~ for RKN a...
 21.3.21.1.56: 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
ISBN: 9780471488859
Solutions for Chapter 21.3: Methods for Systems and Higher Order ODEs
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Chapter 21.3: Methods for Systems and Higher Order ODEs includes 21 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 21 problems in chapter 21.3: Methods for Systems and Higher Order ODEs have been answered, more than 44018 students have viewed full stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859.

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).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Norm
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+ (MoorePenrose 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 = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Skewsymmetric 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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.