 1.2.1P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.2P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.3P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.4P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.5P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.6P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.7P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.8P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.9P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.10P: In Problems, find a function y = f(x) satisfying the given differen...
 1.2.11P: In Problems, find the position function x(t) of a moving particle w...
 1.2.12P: In Problems, find the position function x(t) of a moving particle w...
 1.2.13P: In Problems, find the position function x(t) of a moving particle w...
 1.2.14P: In Problems, find the position function x(t) of a moving particle w...
 1.2.15P: In Problems, find the position function x(t) of a moving particle w...
 1.2.16P: In Problems, find the position function x(t) of a moving particle w...
 1.2.17P: In Problems, find the position function x(t) of a moving particle w...
 1.2.18P: In Problems, find the position function x(t) of a moving particle w...
 1.2.19P: In Problems, a particle starts at the origin and travels along the ...
 1.2.20P: In Problems, a particle starts at the origin and travels along the ...
 1.2.21P: In Problems, a particle starts at the origin and travels along the ...
 1.2.22P: In Problems, a particle starts at the origin and travels along the ...
 1.2.23P: What is the maximum height attained by the arrow of part(b) of Exam...
 1.2.24P: A ball is dropped from the top of a building 400 ft high. How long ...
 1.2.25P: The brakes of a car are applied when it is moving at 100 km/h and p...
 1.2.26P: A projectile is fired straight upward with an initial velocity of 1...
 1.2.27P: A ball is thrown straight downward from the top of a tall building....
 1.2.28P: A baseball is thrown straight downward with an initial speed of 40 ...
 1.2.29P: A diesel car gradually speeds up so that for the first 10 s its acc...
 1.2.30P: A car traveling at 60 mi/h (88 ft/s) skids 176 ft after its brakes ...
 1.2.31P: The skid marks made by an automobile indicated that its brakes were...
 1.2.32P: Suppose that a car skids 15 m if it is moving at 50 km/h when the b...
 1.2.33P: On the planet Gzyx, a ball dropped from a height of 20 ft hits the ...
 1.2.35P: A stone is dropped from rest at an initial height h above the surfa...
 1.2.36P: Suppose a woman has enough “spring” in her legs to jump (on earth) ...
 1.2.37P: At noon a car starts from rest al point A and proceeds at constant ...
 1.2.38P: At noon a car starts from rest at point A and proceeds with constan...
 1.2.39P: If a = 0.5 mi and u0 = 9 mi/h as in Example 4, what must the swimme...
 1.2.40P: Suppose that a = 0.5 mi, u0 = 9 mi/h, and us= 3 mi/h as in Example ...
 1.2.41P: A bomb is dropped from a helicopter hovering at an altitude of 800 ...
 1.2.42P: A spacecraft is in free fall toward the surface of the moon at a sp...
 1.2.43P: Arthur Clarke’s The Wind from the Sun (1963) describes Diana, a spa...
 1.2.44P: A driver involved in an accident claims he was going only 25 mph. W...
Solutions for Chapter 1.2: Differential Equations and Linear Algebra 3rd Edition
Full solutions for Differential Equations and Linear Algebra  3rd Edition
ISBN: 9780136054252
Solutions for Chapter 1.2
Get Full SolutionsChapter 1.2 includes 43 full stepbystep solutions. Differential Equations and Linear Algebra was written by and is associated to the ISBN: 9780136054252. Since 43 problems in chapter 1.2 have been answered, more than 12808 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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.

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

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).