 2.6.1: Consider the system dx dt = x + y dy dt = y. (a) Show that the xax...
 2.6.2: Consider the system dx dt = y dy dt = x + (1 x2)y. (a) Using HPGSys...
 2.6.3: Verify that Y1(t) = (et sin(3t), et cos(3t)) is a solution of this ...
 2.6.4: Verify that Y2(t) = (e(t1) sin(3(t 1)), e(t1) cos(3(t 1))) is a sol...
 2.6.5: Using HPGSystemSolver, sketch the solution curves for Y1(t) and Y2(...
 2.6.6: Recall the Metaphor of the Parking Lot on page 172. Suppose two peo...
 2.6.7: Consider the two drivers, Gib and Harry, from Exercise 6. Suppose t...
 2.6.8: (a) Suppose Y1(t) is a solution of an autonomous system dY/dt = F(Y...
 2.6.9: Suppose Y1(t) and Y2(t) are solutions of an autonomous system dY/dt...
 2.6.10: Consider the system dx dt = 2 dy dt = y2. (a) Calculate the general...
 2.6.11: Consider the system dx dt = x2 + y dy dt = x2 y2. Show that, for th...
Solutions for Chapter 2.6: Existence and Uniqueness for Systems
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 2.6: Existence and Uniqueness for Systems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.6: Existence and Uniqueness for Systems includes 11 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. Since 11 problems in chapter 2.6: Existence and Uniqueness for Systems have been answered, more than 17144 students have viewed full stepbystep solutions from this chapter. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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