- 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 initial-value 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 initial-value problem dy/dt = y, y(0) = 1. Using Euler...
- 1.4.16: Consider the initial-value 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
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 n-l - 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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Invert A by row operations on [A I] to reach [I A-I].
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
lA-II = 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).
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.