 4.1: In 128, find a general solution to the given differential equation
 4.2: In 128, find a general solution to the given differential equation
 4.3: In 128, find a general solution to the given differential equation
 4.4: In 128, find a general solution to the given differential equation
 4.5: In 128, find a general solution to the given differential equation
 4.6: In 128, find a general solution to the given differential equation
 4.7: In 128, find a general solution to the given differential equation
 4.8: In 128, find a general solution to the given differential equation
 4.9: In 128, find a general solution to the given differential equation
 4.10: In 128, find a general solution to the given differential equation
 4.11: In 128, find a general solution to the given differential equation
 4.12: In 128, find a general solution to the given differential equation
 4.13: In 128, find a general solution to the given differential equation
 4.14: In 128, find a general solution to the given differential equation
 4.15: In 128, find a general solution to the given differential equation
 4.16: In 128, find a general solution to the given differential equation
 4.17: In 128, find a general solution to the given differential equation
 4.18: In 128, find a general solution to the given differential equation
 4.19: In 128, find a general solution to the given differential equation
 4.20: In 128, find a general solution to the given differential equation
 4.21: In 128, find a general solution to the given differential equation
 4.22: In 128, find a general solution to the given differential equation
 4.23: In 128, find a general solution to the given differential equation
 4.24: In 128, find a general solution to the given differential equation
 4.25: In 128, find a general solution to the given differential equation
 4.26: In 128, find a general solution to the given differential equation
 4.27: In 128, find a general solution to the given differential equation
 4.28: In 128, find a general solution to the given differential equation
 4.29: In 2936, find the solution to the given initial value problem.
 4.30: In 2936, find the solution to the given initial value problem.
 4.31: In 2936, find the solution to the given initial value problem.
 4.32: In 2936, find the solution to the given initial value problem.
 4.33: In 2936, find the solution to the given initial value problem.
 4.34: In 2936, find the solution to the given initial value problem.
 4.35: In 2936, find the solution to the given initial value problem.
 4.36: In 2936, find the solution to the given initial value problem.
 4.37: Use the massspring oscillator analogy to decide whether all solutio...
 4.38: A 3kg mass is attached to a spring with stiffness k 75 N/m, as in ...
 4.39: A 32lb weight is attached to a vertical spring, causing it to stre...
Solutions for Chapter 4: Fundamentals of Differential Equations and Boundary Value Problems 6th Edition
Full solutions for Fundamentals of Differential Equations and Boundary Value Problems  6th Edition
ISBN: 9780321747747
Solutions for Chapter 4
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Fundamentals of Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780321747747. Since 39 problems in chapter 4 have been answered, more than 2042 students have viewed full stepbystep solutions from this chapter. Chapter 4 includes 39 full stepbystep solutions. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations and Boundary Value Problems, edition: 6.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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.