- 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 Ermentrout-Kopell 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
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.
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 n-l - An).
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.
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.
Invert A by row operations on [A I] to reach [I A-I].
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.
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.