 3.7.1: . In the circuit of Fig. 3.7.7, suppose that L D 5 H, R D 25 , and ...
 3.7.2: Given the same circuit as in 1, suppose that the switch is initiall...
 3.7.3: Suppose that the battery in is replaced with an alternatingcurrent...
 3.7.4: In the circuit of Fig. 3.7.7, with the switch in position 1, suppos...
 3.7.5: In the circuit of Fig. 3.7.7, with the switch in position 1, suppos...
 3.7.6: In the circuit of Fig. 3.7.7, with the switch in position 1, take L...
 3.7.7: (a) Find the charge Q.t / and current I.t / in the RC circuit if E....
 3.7.8: Suppose that in the circuit of Fig. 3.7.8, we have R D 10, C D 0:02...
 3.7.9: Suppose that in the circuit of Fig. 3.7.8, R D 200, C D 2:5 104, Q....
 3.7.10: An emf of voltage E.t / D E0 cos !t is applied to the RC circuit of...
 3.7.11: In 11 through 16, the parameters of an RLC circuit with input volta...
 3.7.12: In 11 through 16, the parameters of an RLC circuit with input volta...
 3.7.13: In 11 through 16, the parameters of an RLC circuit with input volta...
 3.7.14: In 11 through 16, the parameters of an RLC circuit with input volta...
 3.7.15: In 11 through 16, the parameters of an RLC circuit with input volta...
 3.7.16: In 11 through 16, the parameters of an RLC circuit with input volta...
 3.7.17: In 17 through 22, an RLC circuit with input voltage E.t / is descri...
 3.7.18: In 17 through 22, an RLC circuit with input voltage E.t / is descri...
 3.7.19: In 17 through 22, an RLC circuit with input voltage E.t / is descri...
 3.7.20: The circuit and input voltage of with I.0/ D 0
 3.7.21: The circuit and input voltage of with I.0/ D 0 and Q.0/ D 3
 3.7.22: The circuit and input voltage of with I.0/ D 0 and Q.0/ D 0
 3.7.23: Consider an LC circuitthat is, an RLC circuit with R D 0with input ...
 3.7.24: It was stated in the text that, if R, L, and C are positive, then a...
 3.7.25: Prove that the amplitude I0 of the steady periodic solution of Eq. ...
Solutions for Chapter 3.7: Electrical Circuits
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 3.7: Electrical Circuits
Get Full SolutionsDifferential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. Since 25 problems in chapter 3.7: Electrical Circuits have been answered, more than 16579 students have viewed full stepbystep solutions from this chapter. Chapter 3.7: Electrical Circuits includes 25 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
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 nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Factorization
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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + 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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

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.

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.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Solvable system Ax = b.
The right side b is in the column space of A.

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).