 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 28007 students have viewed full stepbystep solutions from this chapter.

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

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.

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).

Column space C (A) =
space of all combinations of the columns of A.

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.

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

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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 .

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.

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.