- 1.2.1: Solve each of the following initial value problems and plot the sol...
- 1.2.2: Follow the instructions for for the following initial value problem...
- 1.2.3: Consider the differential equation dy/dt = ay + b, where both a and...
- 1.2.4: Consider the differential equation dy/dt = ay b. (a) Find the equil...
- 1.2.5: Undetermined Coefficients. Here is an alternative way to solve the ...
- 1.2.6: Use the method of to solve the equation dy/dt = ay + b.
- 1.2.7: The field mouse population in Example 1 satisfies the differential ...
- 1.2.8: Consider a population p of field mice that grows at a rate proporti...
- 1.2.9: The falling object in Example 2 satisfies the initial value problem...
- 1.2.10: Modify Example 2 so that the falling object experiences no air resi...
- 1.2.11: Consider the falling object of mass 10 kg in Example 2, but assume ...
- 1.2.12: A radioactive material, such as the isotope thorium-234, disintegra...
- 1.2.13: The half-life of a radioactive material is the time required for an...
- 1.2.14: Radium-226 has a half-life of 1620 years. Find the time period duri...
- 1.2.15: According to Newtons law of cooling (see of Section 1.1), the tempe...
- 1.2.16: Suppose that a building loses heat in accordance with Newtons law o...
- 1.2.17: Consider an electric circuit containing a capacitor, resistor, and ...
- 1.2.18: A pond containing 1,000,000 gal of water is initially free of a cer...
- 1.2.19: Your swimming pool containing 60,000 gal of water has been contamin...
Solutions for Chapter 1.2: Solutions of Some Differential Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Invert A by row operations on [A I] to reach [I A-I].
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).
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Solvable system Ax = b.
The right side b is in the column space of A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.