- 2.6.1: Consider the system dx dt = x + y dy dt = y. (a) Show that the x-ax...
- 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
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.
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.
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 S-I AS = A = eigenvalue matrix.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Invert A by row operations on [A I] to reach [I A-I].
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 .
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+ (Moore-Penrose 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 ].
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.