 5.1.1: In 14, find the radius of convergence and interval of convergence f...
 5.1.2: In 14, find the radius of convergence and interval of convergence f...
 5.1.3: In 14, find the radius of convergence and interval of convergence f...
 5.1.4: In 14, find the radius of convergence and interval of convergence f...
 5.1.5: In 5 and 6, the given function is analytic at x 0. Find the first f...
 5.1.6: In 5 and 6, the given function is analytic at x 0. Find the first f...
 5.1.7: In 7 and 8, the given function is analytic at x 0. Find the first f...
 5.1.8: In 7 and 8, the given function is analytic at x 0. Find the first f...
 5.1.9: In 9 and 10, rewrite the given power series so that its general ter...
 5.1.10: In 9 and 10, rewrite the given power series so that its general ter...
 5.1.11: In 11 and 12, rewrite the given expression as a single power series...
 5.1.12: In 11 and 12, rewrite the given expression as a single power series...
 5.1.13: In 13 and 14, verify by direct substitution that the given power se...
 5.1.14: In 13 and 14, verify by direct substitution that the given power se...
 5.1.15: In 15 and 16, without actually solving the given differential equat...
 5.1.16: In 15 and 16, without actually solving the given differential equat...
 5.1.17: In 1728, find two power series solutions of the given differential ...
 5.1.18: In 1728, find two power series solutions of the given differential ...
 5.1.19: In 1728, find two power series solutions of the given differential ...
 5.1.20: In 1728, find two power series solutions of the given differential ...
 5.1.21: In 1728, find two power series solutions of the given differential ...
 5.1.22: In 1728, find two power series solutions of the given differential ...
 5.1.23: In 1728, find two power series solutions of the given differential ...
 5.1.24: In 1728, find two power series solutions of the given differential ...
 5.1.25: In 1728, find two power series solutions of the given differential ...
 5.1.26: In 1728, find two power series solutions of the given differential ...
 5.1.27: In 1728, find two power series solutions of the given differential ...
 5.1.28: In 1728, find two power series solutions of the given differential ...
 5.1.29: In 2932, use the power series method to solve the given initialval...
 5.1.30: In 2932, use the power series method to solve the given initialval...
 5.1.31: In 2932, use the power series method to solve the given initialval...
 5.1.32: In 2932, use the power series method to solve the given initialval...
 5.1.33: In 33 and 34, use the procedure in Example 5 to find two power seri...
 5.1.34: In 33 and 34, use the procedure in Example 5 to find two power seri...
 5.1.35: Without actually solving the differential equation (cos x)y y 5y 0,...
 5.1.36: How can the method described in this section be used to find a powe...
 5.1.37: Is x 0 an ordinary or a singular point of the differential equation...
 5.1.38: For purposes of this problem, ignore the graphs given in Figure 5.1...
 5.1.39: (a) Find two power series solutions for y xy y 0 and express the so...
 5.1.40: (a) Find one more nonzero term for each of the solutions y1(x) and ...
Solutions for Chapter 5.1: Solutions about Ordinary Points
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 5.1: Solutions about Ordinary Points
Get Full SolutionsChapter 5.1: Solutions about Ordinary Points includes 40 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Since 40 problems in chapter 5.1: Solutions about Ordinary Points have been answered, more than 39229 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Outer product uv T
= column times row = rank one matrix.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.