 1.3.1: For 18, determine the differential equation giving the slope of the...
 1.3.2: For 18, determine the differential equation giving the slope of the...
 1.3.3: For 18, determine the differential equation giving the slope of the...
 1.3.4: For 18, determine the differential equation giving the slope of the...
 1.3.5: For 18, determine the differential equation giving the slope of the...
 1.3.6: For 18, determine the differential equation giving the slope of the...
 1.3.7: For 18, determine the differential equation giving the slope of the...
 1.3.8: For 18, determine the differential equation giving the slope of the...
 1.3.9: For 912, verify that the given function (or relation) defines a sol...
 1.3.10: For 912, verify that the given function (or relation) defines a sol...
 1.3.11: For 912, verify that the given function (or relation) defines a sol...
 1.3.12: For 912, verify that the given function (or relation) defines a sol...
 1.3.13: Prove that the initialvalue problem y = x sin(x + y), y(0) = 1 has...
 1.3.14: Use the existence and uniqueness theorem to prove that y(x) = 3 is ...
 1.3.15: Do you think that the initialvalue problem y = xy1/2, y(0) = 0 has...
 1.3.16: Even simple looking differential equations can have complicated sol...
 1.3.17: Consider the initialvalue problem: y = y(y 1), y(x0) = y0. (a) Ver...
 1.3.18: For 1821: (a) Determine all equilibrium solutions. (b) Determine th...
 1.3.19: For 1821: (a) Determine all equilibrium solutions. (b) Determine th...
 1.3.20: For 1821: (a) Determine all equilibrium solutions. (b) Determine th...
 1.3.21: For 1821: (a) Determine all equilibrium solutions. (b) Determine th...
 1.3.22: For 2229, sketch the slope field and some representative solution c...
 1.3.23: For 2229, sketch the slope field and some representative solution c...
 1.3.24: For 2229, sketch the slope field and some representative solution c...
 1.3.25: For 2229, sketch the slope field and some representative solution c...
 1.3.26: For 2229, sketch the slope field and some representative solution c...
 1.3.27: For 2229, sketch the slope field and some representative solution c...
 1.3.28: For 2229, sketch the slope field and some representative solution c...
 1.3.29: For 2229, sketch the slope field and some representative solution c...
 1.3.30: According to Newtons law of cooling (see Section 1.1), the temperat...
 1.3.31: For 3136, determine the slope field and some representative solutio...
 1.3.32: For 3136, determine the slope field and some representative solutio...
 1.3.33: For 3136, determine the slope field and some representative solutio...
 1.3.34: For 3136, determine the slope field and some representative solutio...
 1.3.35: For 3136, determine the slope field and some representative solutio...
 1.3.36: For 3136, determine the slope field and some representative solutio...
 1.3.37: (a) Determine the slope field for the differential equation y = x1(...
 1.3.38: Consider the family of curves y = kx2, where k is a constant. (a) S...
 1.3.39: Consider the differential equation di dt + ai = b, where a and b ar...
Solutions for Chapter 1.3: The Geometry of FirstOrder Differential Equations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 1.3: The Geometry of FirstOrder Differential Equations
Get Full SolutionsDifferential Equations was written by and is associated to the ISBN: 9780321964670. Chapter 1.3: The Geometry of FirstOrder Differential Equations includes 39 full stepbystep solutions. Since 39 problems in chapter 1.3: The Geometry of FirstOrder Differential Equations have been answered, more than 21378 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, edition: 4.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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·

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.