 1.4.1: dy dt = 2y + 1, y(0) = 3, 0 t 2, t = 0.5
 1.4.2: dy dt = t y2, y(0) = 1, 0 t 1, t = 0.25
 1.4.3: dy dt = y2 4t, y(0) = 0.5, 0 t 2, t = 0.25
 1.4.4: dy dt = sin y, y(0) = 1, 0 t 3, t = 0.5
 1.4.5: dw dt = (3 w)(w + 1), w(0) = 4, 0 t 5, t = 1.0
 1.4.6: dw dt = (3 w)(w + 1), w(0) = 0, 0 t 5, t = 0.5
 1.4.7: dy dt = e2/y , y(0) = 2, 0 t 2, t = 0.5
 1.4.8: dy dt = e2/y , y(1) = 2, 1 t 3, t = 0.5
 1.4.9: dy dt = y2 y3, y(0) = 0.2, 0 t 10, t = 0.1
 1.4.10: dy dt = 2y3 + t2, y(0) = 0.5, 2 t 2, t = 0.1
 1.4.11: Do a qualitative analysis of the solution of the initialvalue prob...
 1.4.12: As we saw in Exercise 12 of Section 1.1, the velocity v of a freefa...
 1.4.13: Compare your answers to Exercises 7 and 8, and explain your observa...
 1.4.14: Compare your answers to Exercises 5 and 6. Is Eulers method doing a...
 1.4.15: Consider the initialvalue problem dy/dt = y, y(0) = 1. Using Euler...
 1.4.16: Consider the initialvalue problem dy/dt = 2 y, y(0) = 1. Using Eul...
 1.4.17: As we saw in Exercise 19 of Section 1.3, the spiking of a neuron ca...
 1.4.18: vc(0) = 2
 1.4.19: vc(0) = 1
 1.4.20: vc(0) = 1
 1.4.21: vc(0) = 2
Solutions for Chapter 1.4: NUMERICAL TECHNIQUE: EULERS METHOD
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 1.4: NUMERICAL TECHNIQUE: EULERS METHOD
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. Since 21 problems in chapter 1.4: NUMERICAL TECHNIQUE: EULERS METHOD have been answered, more than 16048 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.4: NUMERICAL TECHNIQUE: EULERS METHOD includes 21 full stepbystep solutions.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.