 8.1.1: For 14, find Ly for the given differential operator if (a) y(x) = 2...
 8.1.2: For 14, find Ly for the given differential operator if (a) y(x) = 2...
 8.1.3: For 14, find Ly for the given differential operator if (a) y(x) = 2...
 8.1.4: For 14, find Ly for the given differential operator if (a) y(x) = 2...
 8.1.5: For 59, verify that the given function is in the kernel of L.
 8.1.6: For 59, verify that the given function is in the kernel of L.
 8.1.7: For 59, verify that the given function is in the kernel of L.
 8.1.8: For 59, verify that the given function is in the kernel of L.
 8.1.9: For 59, verify that the given function is in the kernel of L.
 8.1.10: For 1013, compute Ker(L).L = D 3x2.
 8.1.11: For 1013, compute Ker(L).L = D2 + 1.
 8.1.12: For 1013, compute Ker(L).L = D2 + 2D 15. [Hint: Try for two solutio...
 8.1.13: For 1013, compute Ker(L).L = x2D + x.
 8.1.14: For 1415, find L1L2 and L2L1 for the given differential operators, ...
 8.1.15: For 1415, find L1L2 and L2L1 for the given differential operators, ...
 8.1.16: If L1 = D+a1(x), determine all differential operators of the form L...
 8.1.17: For 1718, write the given nonhomogeneous differential equation as a...
 8.1.18: For 1718, write the given nonhomogeneous differential equation as a...
 8.1.19: Use the existence and uniqueness theorem to prove that the only sol...
 8.1.20: Use the existence and uniqueness theorem to formulate and prove a g...
 8.1.21: Determine which of the following sets of vectors is a basis for the...
 8.1.22: Determine which of the following sets of vectors is a basis for the...
 8.1.23: For 2326, determine two linearly independent solutions to the given...
 8.1.24: For 2326, determine two linearly independent solutions to the given...
 8.1.25: For 2326, determine two linearly independent solutions to the given...
 8.1.26: For 2326, determine two linearly independent solutions to the given...
 8.1.27: For 2731, determine three linearly independent solutions to the giv...
 8.1.28: For 2731, determine three linearly independent solutions to the giv...
 8.1.29: For 2731, determine three linearly independent solutions to the giv...
 8.1.30: For 2731, determine three linearly independent solutions to the giv...
 8.1.31: For 2731, determine three linearly independent solutions to the giv...
 8.1.32: For 3233, determine four linearly independent solutions to the give...
 8.1.33: For 3233, determine four linearly independent solutions to the give...
 8.1.34: For 3435, determine two linearly independent solutions to the given...
 8.1.35: For 3435, determine two linearly independent solutions to the given...
 8.1.36: For 3637, determine three linearly independent solutions to the giv...
 8.1.37: For 3637, determine three linearly independent solutions to the giv...
 8.1.38: Determine a particular solution to the given differential equation ...
 8.1.39: Determine a particular solution to the given differential equation ...
 8.1.40: Determine a particular solution to the given differential equation ...
 8.1.41: Determine a particular solution to the given differential equation ...
 8.1.42: Determine a particular solution to the given differential equation ...
 8.1.43: Prove that the linear differential operator of order n, L = Dn + a1...
 8.1.44: Extend the proof of Theorem 8.1.4 to an arbitrary positive integer n.
 8.1.45: Extend the proof of Theorem 8.1.6 to an arbitrary positive integer n.
 8.1.46: Extend the proof of Theorem 8.1.6 to an arbitrary positive integer n.
Solutions for Chapter 8.1: General Theory for Linear Differential Equations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 8.1: General Theory for Linear Differential Equations
Get Full SolutionsChapter 8.1: General Theory for Linear Differential Equations includes 46 full stepbystep solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 46 problems in chapter 8.1: General Theory for Linear Differential Equations have been answered, more than 18938 students have viewed full stepbystep solutions from this chapter.

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!

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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!).

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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 ().

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

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.