 1.10.1: For 15, use Eulers method with the specified step size to determine...
 1.10.2: For 15, use Eulers method with the specified step size to determine...
 1.10.3: For 15, use Eulers method with the specified step size to determine...
 1.10.4: For 15, use Eulers method with the specified step size to determine...
 1.10.5: For 15, use Eulers method with the specified step size to determine...
 1.10.6: For 610, use the modified Euler method with the specified step size...
 1.10.7: For 610, use the modified Euler method with the specified step size...
 1.10.8: For 610, use the modified Euler method with the specified step size...
 1.10.9: For 610, use the modified Euler method with the specified step size...
 1.10.10: For 610, use the modified Euler method with the specified step size...
 1.10.11: For 1115, use the FourthOrder RungeKutta Method with the specifie...
 1.10.12: For 1115, use the FourthOrder RungeKutta Method with the specifie...
 1.10.13: For 1115, use the FourthOrder RungeKutta Method with the specifie...
 1.10.14: For 1115, use the FourthOrder RungeKutta Method with the specifie...
 1.10.15: For 1115, use the FourthOrder RungeKutta Method with the specifie...
 1.10.16: Use the FourthOrder RungeKutta Method with h = 0.5 to approximate...
Solutions for Chapter 1.10: Numerical Solution to FirstOrder Differential Equations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 1.10: Numerical Solution to FirstOrder Differential Equations
Get Full SolutionsSince 16 problems in chapter 1.10: Numerical Solution to FirstOrder Differential Equations have been answered, more than 21401 students have viewed full stepbystep solutions from this chapter. Chapter 1.10: Numerical Solution to FirstOrder Differential Equations includes 16 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. Differential Equations was written by and is associated to the ISBN: 9780321964670. This expansive textbook survival guide covers the following chapters and their solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

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

Outer product uv T
= column times row = rank one matrix.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.