- 1.3.1: dy dt = t 2 + t
- 1.3.2: dy dt = t 2 + 1 3
- 1.3.3: dy dt = 1 2y
- 1.3.4: dy dt = 4y2
- 1.3.5: dy dt = 2y(1 y)
- 1.3.6: dy dt = y + t + 1
- 1.3.7: dy dt = 3y(1 y)
- 1.3.8: dy dt = 2y t
- 1.3.9: dy dt = y + 1 2 (y + t)
- 1.3.10: dy dt = (t + 1)y
- 1.3.11: Suppose we know that the function f (t, y) is continuous and that f...
- 1.3.12: Suppose the constant function y(t) = 2 for all t is a solution of t...
- 1.3.13: Suppose we know that the graph to the right is the graph of the rig...
- 1.3.14: Suppose we know that the graph to the right is the graph of the rig...
- 1.3.15: Consider the autonomous differential equation d S dt = S3 2S2 + S. ...
- 1.3.16: Eight differential equations and four slope fields are given below....
- 1.3.17: Suppose we know that the graph below is the graph of a solution to ...
- 1.3.18: Suppose we know that the graph below is the graph of a solution to ...
- 1.3.19: The spiking of a neuron can be modeled by the differential equation...
- 1.3.20: By separating variables, find the general solution of the different...
- 1.3.21: By separating variables, find the general solution of the different...
- 1.3.22: By separating variables, find the solution of the initial-value pro...
Solutions for Chapter 1.3: QUALITATIVE TECHNIQUE: SLOPE FIELDS
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).
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Upper triangular systems are solved in reverse order Xn to Xl.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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 .
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.