 2.2.41: In each of 1 through 6, find the general solution, using the method...
 2.2.65: This problem gauges the relative effects of initial position and ve...
 2.2.82: In each of 1 through 10, find the general solution x 2 y+ 2x y 6y =...
 2.2.1: In each of 1 through 5, verify that y1 and y2 are solutions of the ...
 2.2.17: In each of 1 through 10, write the general solution. y y 6y = 0 2
 2.2.42: y 4y+ 3y = 2 cos(x + 3) 3
 2.2.66: Repeat the experiment of 1, except now use the critically damped, u...
 2.2.83: x 2 y+ 3x y+ y = 0 3
 2.2.2: y 16y = 0; y(0) = 12, y (0) = 3 y1(x) = e4x , y2(x) = e4x 3
 2.2.18: y 2y+ 10y = 0 3
 2.2.43: y+ 9y = 12 sec(3x)
 2.2.67: 4 through 9 explore the effects of changing the initial position or...
 2.2.84: x 2 y+ x y+ 4y = 0 4
 2.2.3: y+ 3y+ 2y = 0; y(0) = 3, y (0) = 1 y1(x) = e2x , y2(x) = ex 4.
 2.2.19: y+ 6y+ 9y = 0 4
 2.2.44: y 2y 3y = 2 sin2 (x) 5
 2.2.68: y+ 4y+ 2y = 0; y(0) = 0, y (0) = A; A has values 1, 3, 6, 10,4 and ...
 2.2.85: x 2 y+ x y 4y = 0 5
 2.2.4: y 6y+ 13y = 0; y(0) = 1, y (0) = 1 y1(x) = e3x cos(2x), y2(x) = e3x...
 2.2.20: y 3y= 0 5
 2.2.45: y 3y+ 2y = cos(ex ) 6
 2.2.69: y+ 4y+ 4y = 0; y(0) = A, y (0) = 0; A has values 1, 3, 6, 10,4 and ...
 2.2.86: x 2 y+ x y 16y = 0 6
 2.2.5: y 2y+ 2y = 0; y(0) = 6, y (0) = 1 y1(x) = ex cos(x), y2(x) = ex sin...
 2.2.21: y+ 10y+ 26y = 0 6
 2.2.46: y 5y+ 6y = 8 sin2 (4x) I
 2.2.70: y+ 4y+ 4y = 0; y(0) = 0, y (0) = A; A has values 1, 3, 6, 10,4 and ...
 2.2.87: x 2 y+ 3x y+ 10y = 0 7
 2.2.6: y+ 36y = x 1, Yp(x) = (x 1)/36
 2.2.22: y+ 6y 40y = 0 7
 2.2.47: In each of 7 through 16, find the general solution, using the metho...
 2.2.71: y+ 2y+ 5y = 0; y(0) = A, y (0) = 0; A has values 1, 3, 6, 10,4 and ...
 2.2.88: x 2 y+ 6x y+ 6y = 0 8
 2.2.7: y 16y = 4x 2 ; Yp(x) = x 2 /4 + 1/2
 2.2.23: y+ 3y+ 18y = 0 8
 2.2.48: y y 6y = 8e2x 9
 2.2.72: y+ 2y+ 5y = 0; y(0) = 0, y (0) = A; A has values 1, 3, 6, 10,4 and ...
 2.2.89: x 2 y 5x y+ 58y = 0 9
 2.2.8: y+ 3y+ 2y = 15; Yp(x) = 15/2 9
 2.2.24: y+ 16y+ 64y = 0 9
 2.2.49: y 2y+ 10y = 20x 2 + 2x 8 1
 2.2.73: y+ 2y+ 5y = 0; y(0) = 0, y (0) = A; A has values 1, 3, 6, 10,4 and ...
 2.2.90: x 2 y+ 25x y+ 144y = 0 1
 2.2.9: y 6y+ 13y = ex ; Yp(x) = 8ex 1
 2.2.25: y 14y+ 49y = 0 1
 2.2.50: y 4y+ 5y = 21e2x 1
 2.2.74: An object having a mass of 1 gram is attached to the lower end of a...
 2.2.91: x 2 y 11x y+ 35y = 0 I
 2.2.10: y 2y+ 2y = 5x 2 ; Yp(x) = 5x 2 /2 5x 4 1
 2.2.26: y 6y+ 7y = 0 I
 2.2.51: y 6y+ 8y = 3ex 1
 2.2.75: How many times can the mass pass through the equilibrium point in o...
 2.2.92: In each of 11 through 16, solve the initial value problem. x 2 y+ 5...
 2.2.11: Here is a sketch of a proof of Theorem 2.2. Fill in the details. De...
 2.2.27: In each of 11 through 20, solve the initial value problem. y+ 3y= 0...
 2.2.52: y+ 6y+ 9y = 9 cos(3x) 1
 2.2.76: How many times can the mass pass through equilibrium in critical da...
 2.2.93: x 2 y x y= 0; y(2) = 5, y (2) = 8 13
 2.2.12: Let y1(x) = x 2 and y2(x) = x 3 . Show that W(x) = x 4 . Now W(0) =...
 2.2.28: y+ 2y 3y = 0; y(0) = 6, y (0) = 2 13
 2.2.29: y 2y+ y = 0; y(1) = y (1) = 0 14
 2.2.53: y 3y+ 2y = 10 sin(x) 1
 2.2.77: In underdamped, unforced motion, what effect does the damping const...
 2.2.94: x 2 y 3x y+ 4y = 0; y(1) = 4, y (1) = 5 Co
 2.2.13: Show that y1(x) = x and y2(x) = x 2 are linearly independent soluti...
 2.2.30: y 4y+ 4y = 0; y(0) = 3, y (0) = 5 15
 2.2.54: y 4y= 8x 2 + 2e3x 1
 2.2.78: Suppose y(0) = y (0) = 0. Determine the maximum displacement of the...
 2.2.95: x 2 y+ 25x y+ 144y = 0; y(1) = 4, y (1) = 0 15
 2.2.14: Suppose y1 and y2 are solutions of equation (2.2) on (a, b) and tha...
 2.2.31: y+ y 12y = 0; y(2) = 2, y (2) = 1 16
 2.2.55: y 4y+ 13y = 3e2x 5e3x 1
 2.2.79: Consider overdamped forced motion governed by y+ 6y + 2y = 4 cos(3t...
 2.2.96: x 2 y 9x y+ 24y = 0; y(1) = 1, y (1) = 10 16
 2.2.15: Let be a solution of y+ py+ qy = 0 on an open interval I. Suppose (...
 2.2.32: y 2y 5y = 0; y(0) = 0, y (0) = 3 17
 2.2.56: y 2y+ y = 3x + 25 sin(3x) I
 2.2.80: Carry out the program of for the critically damped, forced system g...
 2.2.97: x 2 y+ x y 4y = 0; y(1) = 7, y (1) = 3 17
 2.2.16: Let y1 and y2 be distinct solutions of equation (2.2) on an open in...
 2.2.33: y 2y+ y = 0; y(1) = 12, y (1) = 5 18
 2.2.57: In each of 17 through 24, solve the initial value problem.y 4y = 7e...
 2.2.81: Carry out the program of for the underdamped, forced system governe...
 2.2.98: Here is another approach to solving an Euler equation. For x > 0, s...
 2.2.34: y 5y+ 12y = 0; y(2) = 0, y (2) = 4 19
 2.2.58: y+ 4y= 8 + 34 cos(x); y(0) = 3, y (0) = 2 19
 2.2.99: Outline a procedure for solving the Euler equation for x < 0. Hint:...
 2.2.35: y y+ 4y = 0; y(2) = 1, y (2) = 3 20
 2.2.59: y+ 8y+ 12y = ex + 7; y(0) = 1, y (0) = 0 20
 2.2.36: y+ y y = 0; y(4) = 7, y (4) = 1 21
 2.2.60: y 3y= 2e2x sin(x); y(0) = 1, y (0) = 2 21
 2.2.37: This problem illustrates how small changes in the coefficients of a...
 2.2.61: y 2y 8y = 10ex + 8e2x ; y(0) = 1, y (0) = 4 22
 2.2.38: (a) Find the solution of the initial value problem y 2y + 2 y = 0; ...
 2.2.62: y y+ y = 1; y(1) = 4, y (1) = 2 23
 2.2.39: Suppose is a solution of y+ ay + by = 0; y(x0) = A, y (x0) = B with...
 2.2.63: y y = 5 sin2 (x); y(0) = 2, y (0) = 4 2
 2.2.40: Use power series expansions to derive Eulers formula. Hint: Write e...
 2.2.64: y+ y = tan(x); y(0) = 4, y (0) = 3 2
Solutions for Chapter 2: Linear SecondOrder Equations
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 2: Linear SecondOrder Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Since 99 problems in chapter 2: Linear SecondOrder Equations have been answered, more than 26639 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2: Linear SecondOrder Equations includes 99 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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!).

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.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.