 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 thorium234, disintegra...
 1.2.13: The halflife of a radioactive material is the time required for an...
 1.2.14: Radium226 has a halflife 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
ISBN: 9780470383346
Solutions for Chapter 1.2: Solutions of Some Differential Equations
Get Full SolutionsChapter 1.2: Solutions of Some Differential Equations includes 19 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 19 problems in chapter 1.2: Solutions of Some Differential Equations have been answered, more than 12908 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346.

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.

Complex conjugate
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 SI 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 Fn1c 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!).

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

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.

Normal matrix.
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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Skewsymmetric 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 AI 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)jI 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.