 6.1.6.1.1: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.2: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.3: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.4: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.5: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.6: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.7: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.8: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.9: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.10: In 110 find the interval and radius of convergence for the given po...
 6.1.6.1.11: In 1116 use an appropriate series in (2) to find the Maclaurin seri...
 6.1.6.1.12: In 1116 use an appropriate series in (2) to find the Maclaurin seri...
 6.1.6.1.13: In 1116 use an appropriate series in (2) to find the Maclaurin seri...
 6.1.6.1.14: In 1116 use an appropriate series in (2) to find the Maclaurin seri...
 6.1.6.1.15: In 1116 use an appropriate series in (2) to find the Maclaurin seri...
 6.1.6.1.16: In 1116 use an appropriate series in (2) to find the Maclaurin seri...
 6.1.6.1.17: In 17 and 18 use an appropriate series in (2) to fin the Taylor ser...
 6.1.6.1.18: In 17 and 18 use an appropriate series in (2) to fin the Taylor ser...
 6.1.6.1.19: In 19 and 20 the given function is analytic at Use appropriate seri...
 6.1.6.1.20: In 19 and 20 the given function is analytic at Use appropriate seri...
 6.1.6.1.21: In 21 and 22 the given function is analytic at Use appropriate seri...
 6.1.6.1.22: In 21 and 22 the given function is analytic at Use appropriate seri...
 6.1.6.1.23: In 23 and 24 use a substitution to shift the summation index so tha...
 6.1.6.1.24: In 23 and 24 use a substitution to shift the summation index so tha...
 6.1.6.1.25: In 2530 proceed as in Example 3 to rewrite the given expression usi...
 6.1.6.1.26: In 2530 proceed as in Example 3 to rewrite the given expression usi...
 6.1.6.1.27: In 2530 proceed as in Example 3 to rewrite the given expression usi...
 6.1.6.1.28: In 2530 proceed as in Example 3 to rewrite the given expression usi...
 6.1.6.1.29: In 2530 proceed as in Example 3 to rewrite the given expression usi...
 6.1.6.1.30: In 2530 proceed as in Example 3 to rewrite the given expression usi...
 6.1.6.1.31: In 3134 verify by direct substitution that the given power series i...
 6.1.6.1.32: In 3134 verify by direct substitution that the given power series i...
 6.1.6.1.33: In 3134 verify by direct substitution that the given power series i...
 6.1.6.1.34: In 3134 verify by direct substitution that the given power series i...
 6.1.6.1.35: In 3538 proceed as in Example 4 and find a power series solution of...
 6.1.6.1.36: In 3538 proceed as in Example 4 and find a power series solution of...
 6.1.6.1.37: In 3538 proceed as in Example 4 and find a power series solution of...
 6.1.6.1.38: In 3538 proceed as in Example 4 and find a power series solution of...
 6.1.6.1.39: In 19, find an easier way than multiplying two power series to obta...
 6.1.6.1.40: In 21, what do you think is the interval of convergence for the Mac...
Solutions for Chapter 6.1: Series Solutions of Linear Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 6.1: Series Solutions of Linear Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Since 40 problems in chapter 6.1: Series Solutions of Linear Equations have been answered, more than 20423 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Chapter 6.1: Series Solutions of Linear Equations includes 40 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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!

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.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.