 1.2.1: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.2: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.3: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.4: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.5: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.6: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.7: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.8: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.9: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.10: In 1 through 10, find a function y D f .x/ satisfying the given dif...
 1.2.11: In 11 through 18, find the position function x.t / of a moving part...
 1.2.12: In 11 through 18, find the position function x.t / of a moving part...
 1.2.13: In 11 through 18, find the position function x.t / of a moving part...
 1.2.14: In 11 through 18, find the position function x.t / of a moving part...
 1.2.15: In 11 through 18, find the position function x.t / of a moving part...
 1.2.16: In 11 through 18, find the position function x.t / of a moving part...
 1.2.17: In 11 through 18, find the position function x.t / of a moving part...
 1.2.18: In 11 through 18, find the position function x.t / of a moving part...
 1.2.19: In 19 through 22, a particle starts at the origin and travels along...
 1.2.20: In 19 through 22, a particle starts at the origin and travels along...
 1.2.21: In 19 through 22, a particle starts at the origin and travels along...
 1.2.22: In 19 through 22, a particle starts at the origin and travels along...
 1.2.23: What is the maximum height attained by the arrow of part (b) of Exa...
 1.2.24: A ball is dropped from the top of a building 400 ft high. How long ...
 1.2.25: The brakes of a car are applied when it is moving at 100 km=h and p...
 1.2.26: A projectile is fired straight upward with an initial velocity of 1...
 1.2.27: A ball is thrown straight downward from the top of a tall building....
 1.2.28: A baseball is thrown straight downward with an initial speed of 40 ...
 1.2.29: A baseball is thrown straight downward with an initial speed of 40 ...
 1.2.30: A car traveling at 60 mi=h (88 ft=s) skids 176 ft after its brakes ...
 1.2.31: The skid marks made by an automobile indicated that its brakes were...
 1.2.32: Suppose that a car skids 15 m if it is moving at 50 km=h when the b...
 1.2.33: On the planet Gzyx, a ball dropped from a height of 20 ft hits the ...
 1.2.34: A person can throw a ball straight upward from the surface of the e...
 1.2.35: A stone is dropped from rest at an initial height h above the surfa...
 1.2.36: Suppose a woman has enough spring in her legs to jump (on earth) fr...
 1.2.37: At noon a car starts from rest at point A and proceeds at constant ...
 1.2.38: At noon a car starts from rest at point A and proceeds with constan...
 1.2.39: If a D 0:5 mi and v0 D 9 mi=h as in Example 4, what must the swimme...
 1.2.40: Suppose that a D 0:5 mi, v0 D 9 mi=h, and vS D 3 mi=h as in Example...
 1.2.41: A bomb is dropped from a helicopter hovering at an altitude of 800 ...
 1.2.42: A spacecraft is in free fall toward the surface of the moon at a sp...
 1.2.43: Arthur Clarkes The Wind from the Sun (1963) describes Diana, a spac...
 1.2.44: A driver involved in an accident claims he was going only 25 mph. W...
Solutions for Chapter 1.2: Integrals as General and Particular Solutions
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 1.2: Integrals as General and Particular Solutions
Get Full SolutionsDifferential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.2: Integrals as General and Particular Solutions includes 44 full stepbystep solutions. Since 44 problems in chapter 1.2: Integrals as General and Particular Solutions have been answered, more than 15232 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Outer product uv T
= column times row = rank one matrix.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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