 4.1.1: In each of 1 through 6, determine intervals in which solutions are ...
 4.1.2: In each of 1 through 6, determine intervals in which solutions are ...
 4.1.3: In each of 1 through 6, determine intervals in which solutions are ...
 4.1.4: In each of 1 through 6, determine intervals in which solutions are ...
 4.1.5: In each of 1 through 6, determine intervals in which solutions are ...
 4.1.6: In each of 1 through 6, determine intervals in which solutions are ...
 4.1.7: In each of 7 through 10, determine whether the given functions are ...
 4.1.8: In each of 7 through 10, determine whether the given functions are ...
 4.1.9: In each of 7 through 10, determine whether the given functions are ...
 4.1.10: In each of 7 through 10, determine whether the given functions are ...
 4.1.11: In each of 11 through 16, verify that the given functions are solut...
 4.1.12: In each of 11 through 16, verify that the given functions are solut...
 4.1.13: In each of 11 through 16, verify that the given functions are solut...
 4.1.14: In each of 11 through 16, verify that the given functions are solut...
 4.1.15: In each of 11 through 16, verify that the given functions are solut...
 4.1.16: In each of 11 through 16, verify that the given functions are solut...
 4.1.17: Show that W(5, sin2 t, cos 2t) = 0 for all t. Can you establish thi...
 4.1.18: Verify that the differential operator defined byL[y] = y(n) + p1(t)...
 4.1.19: Let the linear differential operator L be defined byL[y] = a0y(n) +...
 4.1.20: In this problem we show how to generalize Theorem 3.2.7 (Abels theo...
 4.1.21: In each of 21 through 24, use Abels formula ( 20) to find the Wrons...
 4.1.22: In each of 21 through 24, use Abels formula ( 20) to find the Wrons...
 4.1.23: In each of 21 through 24, use Abels formula ( 20) to find the Wrons...
 4.1.24: In each of 21 through 24, use Abels formula ( 20) to find the Wrons...
 4.1.25: (a) Show that the functions f(t) = t2t and g(t) = t3 are linearly...
 4.1.26: Show that if y1 is a solution ofy + p1(t)y + p2(t)y + p3(t)y = 0,th...
 4.1.27: In each of 27 and 28, use the method of reduction of order ( 26) to...
 4.1.28: In each of 27 and 28, use the method of reduction of order ( 26) to...
Solutions for Chapter 4.1: General Theory of nth Order Linear Equations
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 4.1: General Theory of nth Order Linear Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Chapter 4.1: General Theory of nth Order Linear Equations includes 28 full stepbystep solutions. Since 28 problems in chapter 4.1: General Theory of nth Order Linear Equations have been answered, more than 11384 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

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.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

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 ().

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.