 4.1.4.1.1: In 14 the given family of functions is the general solution of the ...
 4.1.4.1.2: In 14 the given family of functions is the general solution of the ...
 4.1.4.1.3: In 14 the given family of functions is the general solution of the ...
 4.1.4.1.4: In 14 the given family of functions is the general solution of the ...
 4.1.4.1.5: Given that y c1 c2x2 is a twoparameter family of solutions of xy y...
 4.1.4.1.6: Find two members of the family of solutions in that satisfy the ini...
 4.1.4.1.7: Given that x(t) c1 cos vt c2 sin vt is the general solution of x v2...
 4.1.4.1.8: Use the general solution of x v2x 0 given in to show that a solutio...
 4.1.4.1.9: In 9 and 10 find an interval centered about x 0 for which the given...
 4.1.4.1.10: In 9 and 10 find an interval centered about x 0 for which the given...
 4.1.4.1.11: (a) Use the family in to find a solution of y y 0 that satisfies th...
 4.1.4.1.12: Use the family in to find a solution of xy y 0 that satisfies the b...
 4.1.4.1.13: In 13 and 14 the given twoparameter family is a solution of the in...
 4.1.4.1.14: In 13 and 14 the given twoparameter family is a solution of the in...
 4.1.4.1.15: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.16: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.17: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.18: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.19: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.20: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.21: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.22: In 1522 determine whether the given set of functions is linearly in...
 4.1.4.1.23: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.24: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.25: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.26: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.27: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.28: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.29: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.30: In 2330 verify that the given functions form a fundamental set of s...
 4.1.4.1.31: In 3134 verify that the given twoparameter family of functions is ...
 4.1.4.1.32: In 3134 verify that the given twoparameter family of functions is ...
 4.1.4.1.33: In 3134 verify that the given twoparameter family of functions is ...
 4.1.4.1.34: In 3134 verify that the given twoparameter family of functions is ...
 4.1.4.1.35: (a) Verify that and are, respectively, particular solutions of and ...
 4.1.4.1.36: (a) By inspection find a particular solution of y 2y 10. (b) By ins...
 4.1.4.1.37: Let n 1, 2, 3, . . . . Discuss how the observations Dn x n1 0 and D...
 4.1.4.1.38: Suppose that y1 ex and y2 ex are two solutions of a homogeneous lin...
 4.1.4.1.39: (a) Verify that y1 x3 and y2 x 3 are linearly independent solutions...
 4.1.4.1.40: Is the set of functions f1(x) ex2 , f2(x) ex3 linearly dependent or...
 4.1.4.1.41: Suppose y1, y2, . . . , yk are k linearly independent solutions on ...
 4.1.4.1.42: Suppose that y1, y2, . . . , yk are k nontrivial solutions of a hom...
 4.1.4.1.412: In 1 and 2 verify that y1 and y2 are solutions of the given differe...
 4.1.4.1.413: In 1 and 2 verify that y1 and y2 are solutions of the given differe...
 4.1.4.1.414: In 38 solve the given differential equation by using the substituti...
 4.1.4.1.415: In 38 solve the given differential equation by using the substituti...
 4.1.4.1.416: In 38 solve the given differential equation by using the substituti...
 4.1.4.1.417: In 38 solve the given differential equation by using the substituti...
 4.1.4.1.418: In 38 solve the given differential equation by using the substituti...
 4.1.4.1.419: In 38 solve the given differential equation by using the substituti...
 4.1.4.1.420: In 9 and 10 solve the given initialvalue problem.2yy 1, y(0) 2, y(...
 4.1.4.1.421: In 9 and 10 solve the given initialvalue problem.y x(y)2 0, y(1) 4...
 4.1.4.1.422: Consider the initialvalue problem y yy 0, y(0) 1, y(0) 1. (a) Use ...
 4.1.4.1.423: Find two solutions of the initialvalue problem Use a numerical sol...
 4.1.4.1.424: In 13 and 14 show that the substitution u y leads to a Bernoulli eq...
 4.1.4.1.425: In 13 and 14 show that the substitution u y leads to a Bernoulli eq...
 4.1.4.1.426: In 1518 proceed as in Example 3 and obtain the first six nonzero te...
 4.1.4.1.427: In 1518 proceed as in Example 3 and obtain the first six nonzero te...
 4.1.4.1.428: In 1518 proceed as in Example 3 and obtain the first six nonzero te...
 4.1.4.1.429: In 1518 proceed as in Example 3 and obtain the first six nonzero te...
 4.1.4.1.430: In calculus the curvature of a curve that is defined by a function ...
 4.1.4.1.431: In we saw that cos x and ex were solutions of the nonlinear equatio...
 4.1.4.1.432: Discuss how the method of reduction of order considered in this sec...
 4.1.4.1.433: Discuss how to find an alternative twoparameter family of solution...
 4.1.4.1.434: Motion in a Force Field A mathematical model for the position x(t) ...
 4.1.4.1.435: A mathematical model for the position x(t) of a moving object is . ...
Solutions for Chapter 4.1: HigherOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 4.1: HigherOrder Differential Equations
Get Full SolutionsChapter 4.1: HigherOrder Differential Equations includes 66 full stepbystep solutions. Since 66 problems in chapter 4.1: HigherOrder Differential Equations have been answered, more than 22599 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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 ().

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.