 6.5.1: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.2: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.3: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.4: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.5: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.6: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.7: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.8: In each of 1 through 8: a. Find the solution of the given initial v...
 6.5.9: Consider again the system in Example 1 of this section, in which an...
 6.5.10: Consider the initial value problem y__ + y_ + y = (t 1), y(0) = 0, ...
 6.5.11: Consider the initial value problem y__ + y_ + y = k (t 1), y(0) = 0...
 6.5.12: Consider the initial value problem y__ + y = fk (t), y(0) = 0, y_(0...
 6.5.13: 13 through 16 deal with the effect of a sequence of impulses on an ...
 6.5.14: 13 through 16 deal with the effect of a sequence of impulses on an ...
 6.5.15: 13 through 16 deal with the effect of a sequence of impulses on an ...
 6.5.16: 13 through 16 deal with the effect of a sequence of impulses on an ...
 6.5.17: The position of a certain lightly damped oscillator satisfies the i...
 6.5.18: Proceed as in for the oscillator satisfying y__ + 0.1y_ + y = 15 _ ...
 6.5.19: a. By the method of variation of parameters, show that the solution...
Solutions for Chapter 6.5: Impulse Functions
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 6.5: Impulse Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 19 problems in chapter 6.5: Impulse Functions have been answered, more than 13720 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Chapter 6.5: Impulse Functions includes 19 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11.

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!

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

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

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

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 inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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(DÂ» O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).