 7.1.1: In each of 1 through 4, transform the given equation into a system ...
 7.1.2: In each of 1 through 4, transform the given equation into a system ...
 7.1.3: In each of 1 through 4, transform the given equation into a system ...
 7.1.4: In each of 1 through 4, transform the given equation into a system ...
 7.1.5: In each of 5 and 6, transform the given initial value problem into ...
 7.1.6: In each of 5 and 6, transform the given initial value problem into ...
 7.1.7: Systems of first order equations can sometimes be transformed into ...
 7.1.8: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
 7.1.9: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
 7.1.10: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
 7.1.11: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
 7.1.12: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
 7.1.13: Transform Eqs. (2) for the parallel circuit into a single second or...
 7.1.14: Show that if a11, a12, a21, and a22 are constants with a12 and a21 ...
 7.1.15: Consider the linear homogeneous systemx = p11(t)x + p12(t)y,y = p21...
 7.1.16: Let x = x1(t), y = y1(t) and x = x2(t), y = y2(t) be any two soluti...
 7.1.17: Equations (1) can be derived by drawing a freebody diagram showing...
 7.1.18: Transform the system (1) into a system of first order equations by ...
 7.1.19: Electric Circuits. The theory of electric circuits, such as that sh...
 7.1.20: Electric Circuits. The theory of electric circuits, such as that sh...
 7.1.21: Electric Circuits. The theory of electric circuits, such as that sh...
 7.1.22: Consider the two interconnected tanks shown in Figure 7.1.6. Tank 1...
 7.1.23: Consider two interconnected tanks similar to those in Figure 7.1.6....
Solutions for Chapter 7.1: Introduction
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 7.1: Introduction
Get Full SolutionsSince 23 problems in chapter 7.1: Introduction have been answered, more than 12196 students have viewed full stepbystep solutions from this chapter. Chapter 7.1: Introduction includes 23 full stepbystep solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.