 1.1.1: In each of 1 through 4, draw a direction field for the given differ...
 1.1.2: In each of 1 through 4, draw a direction field for the given differ...
 1.1.3: In each of 1 through 4, draw a direction field for the given differ...
 1.1.4: In each of 1 through 4, draw a direction field for the given differ...
 1.1.5: In each of 5 and 6, write down a differential equation of the form ...
 1.1.6: In each of 5 and 6, write down a differential equation of the form ...
 1.1.7: In each of 7 through 10, draw a direction field for the given diffe...
 1.1.8: In each of 7 through 10, draw a direction field for the given diffe...
 1.1.9: In each of 7 through 10, draw a direction field for the given diffe...
 1.1.10: In each of 7 through 10, draw a direction field for the given diffe...
 1.1.11: The direction field of Figure 1.1.5.
 1.1.12: The direction field of Figure 1.1.6
 1.1.13: The direction field of Figure 1.1.7
 1.1.14: The direction field of Figure 1.1.8.
 1.1.15: The direction field of Figure 1.1.9.
 1.1.16: The direction field of Figure 1.1.10.
 1.1.17: Apond initially contains 1,000,000 gal of water and an unknown amou...
 1.1.18: A spherical raindrop evaporates at a rate proportional to its surfa...
 1.1.19: Newtons law of cooling states that the temperature of an object cha...
 1.1.20: A certain drug is being administered intravenously to a hospital pa...
 1.1.21: For small, slowly falling objects, the assumption made in the text ...
 1.1.22: In each of 22 through 25, draw a direction field for the given diff...
 1.1.23: In each of 22 through 25, draw a direction field for the given diff...
 1.1.24: In each of 22 through 25, draw a direction field for the given diff...
 1.1.25: In each of 22 through 25, draw a direction field for the given diff...
Solutions for Chapter 1.1: Some Basic Mathematical Models; Direction Fields
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 1.1: Some Basic Mathematical Models; Direction Fields
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Since 25 problems in chapter 1.1: Some Basic Mathematical Models; Direction Fields have been answered, more than 12609 students have viewed full stepbystep solutions from this chapter. Chapter 1.1: Some Basic Mathematical Models; Direction Fields includes 25 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.