 4.4.1: In each of 1 through 6 use the method of variation of parameters to...
 4.4.2: In each of 1 through 6 use the method of variation of parameters to...
 4.4.3: In each of 1 through 6 use the method of variation of parameters to...
 4.4.4: In each of 1 through 6 use the method of variation of parameters to...
 4.4.5: In each of 1 through 6 use the method of variation of parameters to...
 4.4.6: In each of 1 through 6 use the method of variation of parameters to...
 4.4.7: In each of 7 and 8 find the general solution of the given different...
 4.4.8: In each of 7 and 8 find the general solution of the given different...
 4.4.9: In each of 9 through 12 find the solution of the given initial valu...
 4.4.10: In each of 9 through 12 find the solution of the given initial valu...
 4.4.11: In each of 9 through 12 find the solution of the given initial valu...
 4.4.12: In each of 9 through 12 find the solution of the given initial valu...
 4.4.13: Given that x, x2, and 1/x are solutions of the homogeneous equation...
 4.4.14: Find a formula involving integrals for a particular solution of the...
 4.4.15: Find a formula involving integrals for a particular solution of the...
 4.4.16: Find a formula involving integrals for a particular solution of the...
 4.4.17: Find a formula involving integrals for a particular solution of the...
Solutions for Chapter 4.4: The Method of Variation of Parameters
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 4.4: The Method of Variation of Parameters
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Since 17 problems in chapter 4.4: The Method of Variation of Parameters have been answered, more than 12726 students have viewed full stepbystep solutions from this chapter. Chapter 4.4: The Method of Variation of Parameters includes 17 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

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

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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