 1.3.1: In each of 1 through 6, determine the order of the given differenti...
 1.3.2: In each of 1 through 6, determine the order of the given differenti...
 1.3.3: In each of 1 through 6, determine the order of the given differenti...
 1.3.4: In each of 1 through 6, determine the order of the given differenti...
 1.3.5: In each of 1 through 6, determine the order of the given differenti...
 1.3.6: In each of 1 through 6, determine the order of the given differenti...
 1.3.7: In each of 7 through 14, verify that each given function is a solut...
 1.3.8: In each of 7 through 14, verify that each given function is a solut...
 1.3.9: In each of 7 through 14, verify that each given function is a solut...
 1.3.10: In each of 7 through 14, verify that each given function is a solut...
 1.3.11: In each of 7 through 14, verify that each given function is a solut...
 1.3.12: In each of 7 through 14, verify that each given function is a solut...
 1.3.13: In each of 7 through 14, verify that each given function is a solut...
 1.3.14: In each of 7 through 14, verify that each given function is a solut...
 1.3.15: In each of 15 through 18, determine the values of r for which the g...
 1.3.16: In each of 15 through 18, determine the values of r for which the g...
 1.3.17: In each of 15 through 18, determine the values of r for which the g...
 1.3.18: In each of 15 through 18, determine the values of r for which the g...
 1.3.19: In each of 19 and 20, determine the values of r for which the given...
 1.3.20: In each of 19 and 20, determine the values of r for which the given...
 1.3.21: In each of 21 through 24, determine the order of the given partial ...
 1.3.22: In each of 21 through 24, determine the order of the given partial ...
 1.3.23: In each of 21 through 24, determine the order of the given partial ...
 1.3.24: In each of 21 through 24, determine the order of the given partial ...
 1.3.25: In each of 25 through 28, verify that each given function is a solu...
 1.3.26: In each of 25 through 28, verify that each given function is a solu...
 1.3.27: In each of 25 through 28, verify that each given function is a solu...
 1.3.28: In each of 25 through 28, verify that each given function is a solu...
 1.3.29: Follow the steps indicated here to derive the equation of motion of...
 1.3.30: Another way to derive the pendulum equation (12) is based on the pr...
 1.3.31: A third derivation of the pendulum equation depends on the principl...
Solutions for Chapter 1.3: Classification of Differential Equations
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 1.3: Classification of Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Chapter 1.3: Classification of Differential Equations includes 31 full stepbystep solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. This expansive textbook survival guide covers the following chapters and their solutions. Since 31 problems in chapter 1.3: Classification of Differential Equations have been answered, more than 9378 students have viewed full stepbystep solutions from this chapter.

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.

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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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.

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

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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