- 6.1.1: f (t) = 3 (a constant function)
- 6.1.2: g(t) = t
- 6.1.3: h(t) = 5t2
- 6.1.4: k(t) = t5
- 6.1.5: Verify that L[t n] = n! sn+1 (s > 0). [Hint: A rigorous derivation ...
- 6.1.6: Using L[t n] = n! sn+1 (s > 0), give a formula for the Laplace tran...
- 6.1.7: 1 s 3
- 6.1.8: 5 3s
- 6.1.9: 2 3s + 5
- 6.1.10: 14 (3s + 2)(s 4)
- 6.1.11: 4 s(s + 3)
- 6.1.12: 5 (s 1)(s 2)
- 6.1.13: 2s + 1 (s 1)(s 2)
- 6.1.14: 2s2 + 3s 2 s(s + 1)(s 2)
- 6.1.15: dy dt = y + e2t , y(0) = 2
- 6.1.16: dy dt + 5y = et , y(0) = 2
- 6.1.17: dy dt + 7y = 1, y(0) = 3
- 6.1.18: dy dt + 4y = 6, y(0) = 0
- 6.1.19: dy dt + 9y = 2, y(0) = 2
- 6.1.20: dy dt = y + 2, y(0) = 4
- 6.1.21: dy dt = y + e2t , y(0) = 1
- 6.1.22: dy dt = 2y + t, y(0) = 0
- 6.1.23: dy dt = y + t2, y(0) = 1
- 6.1.24: dy dt + 4y = 2 + 3t, y(0) = 1
- 6.1.25: Find the general solution of the equation dy dt = 2y + 2e3t . (This...
- 6.1.26: Suppose g(t) = f (t) dt; that is, g(t) is the antiderivative of f (...
- 6.1.27: All of the examples in this section and all the differential equati...
Solutions for Chapter 6.1: LAPLACE TRANSFORMS
Full solutions for Differential Equations 00 | 4th Edition
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
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.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
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.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.