 1.6.1: dy dt = 3y(y 2)
 1.6.2: dy dt = y2 4y 12
 1.6.3: dy dt = cos y
 1.6.4: dw dt = w cos w
 1.6.5: dw dt = (1 w)sin w
 1.6.6: dy dt = 1 y 2
 1.6.7: dv dt = v2 2v 2
 1.6.8: dw dt = 3w3 12w2
 1.6.9: dy dt = 1 + cos y
 1.6.10: dy dt = tan y
 1.6.11: dy dt = y ln y
 1.6.12: dw dt = (w22) arctan w
 1.6.13: Equation from Exercise 1; y(0) = 1, y(2) = 1, y(0) = 3, y(0) = 2.
 1.6.14: Equation from Exercise 2; y(0) = 1, y(1) = 0, y(0) = 6, y(0) = 5.
 1.6.15: Equation from Exercise 3; y(0) = 0, y(1) = 1, y(0) = /2, y(0) = .
 1.6.16: Equation from Exercise 4; w(0) = 0, w(3) = 1, w(0) = 2, w(0) = 1.
 1.6.17: Equation from Exercise 5; w(0) = 3/2, w(0) = 1, w(0) = 2, w(0) = 3.
 1.6.18: Equation from Exercise 6; y(0) = 0, y(1) = 3, y(0) = 2 (trick quest...
 1.6.19: Equation from Exercise 7; v(0) = 0, v(1) = 1, v(0) = 1.
 1.6.20: Equation from Exercise 8; w(0) = 1, w(0) = 0, w(0) = 3, w(1) = 3.
 1.6.21: Equation from Exercise 9; y(0) = , y(0) = 0, y(0) = , y(0) = 2.
 1.6.22: y(0) = 1
 1.6.23: y(0) = 2
 1.6.24: y(0) = 2
 1.6.25: y(0) = 4
 1.6.26: y(0) = 4
 1.6.27: y(3) = 1
 1.6.28: Consider the autonomous equation dy/dt = f (y) where f (y) is conti...
 1.6.29: In Exercises 2932, the graph of a function f (y) is given. Sketch t...
 1.6.30: In Exercises 2932, the graph of a function f (y) is given. Sketch t...
 1.6.31: In Exercises 2932, the graph of a function f (y) is given. Sketch t...
 1.6.32: In Exercises 2932, the graph of a function f (y) is given. Sketch t...
 1.6.33: In Exercises 3336, a phase line for an autonomous equation dy/dt = ...
 1.6.34: In Exercises 3336, a phase line for an autonomous equation dy/dt = ...
 1.6.35: In Exercises 3336, a phase line for an autonomous equation dy/dt = ...
 1.6.36: In Exercises 3336, a phase line for an autonomous equation dy/dt = ...
 1.6.37: Eight differential equations and four phase lines are given below. ...
 1.6.38: Let f (y) be a continuous function. (a) Suppose that f (10) > 0 and...
 1.6.39: Suppose you wish to model a population with a differential equation...
 1.6.40: Consider the ErmentroutKopell model for the spiking of a neuron d ...
 1.6.41: Use PhaseLines to describe the phase line for the differential equa...
 1.6.42: Use PhaseLines to describe the phase line for the differential equa...
 1.6.43: Suppose dy/dt = f (y) has an equilibrium point at y = y0 and (a) f ...
 1.6.44: (a) Sketch the phase line for the differential equation dy dt = 1 (...
 1.6.45: Let x(t) be the amount of time between two consecutive trolley cars...
 1.6.46: For the model in Exercise 45: (a) Find the equilibrium points. (b) ...
 1.6.47: Use the model in Exercise 45 to predict what happens to x(t) as t i...
 1.6.48: Assuming the model for x(t) from Exercise 45, what happens if troll...
Solutions for Chapter 1.6: EQUILIBRIA AND THE PHASE LINE
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 1.6: EQUILIBRIA AND THE PHASE LINE
Get Full SolutionsDifferential Equations 00 was written by and is associated to the ISBN: 9780495561989. Chapter 1.6: EQUILIBRIA AND THE PHASE LINE includes 48 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. Since 48 problems in chapter 1.6: EQUILIBRIA AND THE PHASE LINE have been answered, more than 16122 students have viewed full stepbystep solutions from this chapter.

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.

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.

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).

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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.

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

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 .

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.

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

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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