- 1.2.1: Bob, Glen, and Paul are once again sitting around enjoying their ni...
- 1.2.2: Make up a differential equation of the form dy dt = 2y t + g(y) tha...
- 1.2.3: Make up a differential equation of the form dy/dt = f (t, y) that h...
- 1.2.4: In Section 1.1, we guessed solutions to the exponential growth mode...
- 1.2.5: dy dt = (ty) 2
- 1.2.6: dy dt = t 4 y
- 1.2.7: dy dt = 2y + 1
- 1.2.8: dy dt = 2 y
- 1.2.9: dy dt = ey
- 1.2.10: dx dt = 1 + x2
- 1.2.11: dy dt = 2ty2 + 3y2
- 1.2.12: dy dt = t y
- 1.2.13: dy dt = t t2 y + y
- 1.2.14: dy dt = t 3 y
- 1.2.15: dy dt = 1 2y + 1
- 1.2.16: dy dt = 2y + 1 t
- 1.2.17: dy dt = y(1 y)
- 1.2.18: dy dt = 4t 1 + 3y2
- 1.2.19: dv dt = t 2v 2 2v + t 2
- 1.2.20: dy dt = 1 ty + t + y + 1
- 1.2.21: dy dt = et y 1 + y2
- 1.2.22: dy dt = y2 4
- 1.2.23: dw dt = w t
- 1.2.24: dy dx = sec y
- 1.2.25: dx dt = xt, x(0) = 1/
- 1.2.26: dy dt = ty, y(0) = 3
- 1.2.27: dy dt = y2, y(0) = 1/2
- 1.2.28: dy dt = t 2 y3, y(0) = 1
- 1.2.29: dy dt = y2, y(0) = 0
- 1.2.30: dy dt = t y t2 y , y(0) = 4
- 1.2.31: dy dt = 2y + 1, y(0) = 3
- 1.2.32: dy dt = ty2 + 2y2, y(0) = 1
- 1.2.33: dx dt = t2 x + t3x , x(0) = 2
- 1.2.34: dy dt = 1 y2 y , y(0) = 2
- 1.2.35: dy dt = (y2 + 1)t, y(0) = 1
- 1.2.36: dy dt = 1 2y + 3 , y(0) = 1
- 1.2.37: dy dt = 2ty2 + 3t 2 y2, y(1) = 1
- 1.2.38: dy dt = y2 + 5 y , y(0) = 2
- 1.2.39: A 5-gallon bucket is full of pure water. Suppose we begin dumping s...
- 1.2.40: Consider the following very simple model of blood cholesterol level...
- 1.2.41: A cup of hot chocolate is initially 170 F and is left in a room wit...
- 1.2.42: Suppose you are having a dinner party for a large group of people, ...
- 1.2.43: In Exercise 12 of Section 1.1, we saw that the velocity v of a free...
Solutions for Chapter 1.2: ANALYTIC TECHNIQUE: SEPARATION OF VARIABLES
Full solutions for Differential Equations 00 | 4th Edition
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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 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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
A directed graph that has constants Cl, ... , Cm associated with the edges.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.