 6.2.1: Use the RK4 method with h 0.1 to approximate y(0.5), where y(x) is ...
 6.2.2: Assume that w2 3 4 in (4). Use the resulting secondorder RungeKutt...
 6.2.3: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.4: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.5: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.6: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.7: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.8: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.9: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.10: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.11: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.12: In 312, use the RK4 method with h 0.1 to obtain a fourdecimal appr...
 6.2.13: If air resistance is proportional to the square of the instantaneou...
 6.2.14: A mathematical model for the area A (in cm2 ) that a colony of bact...
 6.2.15: Consider the initialvalue problem y x 2 y 3 , y(1) 1. See in Exerc...
 6.2.16: Consider the initialvalue problem y 2y, y(0) 1. The analytic solut...
 6.2.17: Repeat using the initialvalue problem y 2y x, y(0) 1. The analytic...
 6.2.18: Consider the initialvalue problem y 2x 3y 1, y(1) 5. The analytic ...
 6.2.19: Repeat for the initialvalue problem y ey , y(0) 0. The analytic so...
 6.2.20: A count of the number of evaluations of the function f used in solv...
 6.2.21: The RK4 method for solving an initialvalue problem over an interva...
Solutions for Chapter 6.2: RungeKutta Methods
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 6.2: RungeKutta Methods
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 21 problems in chapter 6.2: RungeKutta Methods have been answered, more than 35008 students have viewed full stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Chapter 6.2: RungeKutta Methods includes 21 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).