 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 and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 1.3: Classification of Differential Equations
Get Full SolutionsThis 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 16332 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: 9780470458310. Chapter 1.3: Classification of Differential Equations includes 31 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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)·(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 .

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).