 9.3.1: Find all real solutions of the differential equations in Exercises ...
 9.3.2: Find all real solutions of the differential equations in Exercises ...
 9.3.3: Find all real solutions of the differential equations in Exercises ...
 9.3.4: Find all real solutions of the differential equations in Exercises ...
 9.3.5: Find all real solutions of the differential equations in Exercises ...
 9.3.6: Find all real solutions of the differential equations in Exercises ...
 9.3.7: Find all real solutions of the differential equations in Exercises ...
 9.3.8: Find all real solutions of the differential equations in Exercises ...
 9.3.9: Find all real solutions of the differential equations in Exercises ...
 9.3.10: Find all real solutions of the differential equations in Exercises ...
 9.3.11: Find all real solutions of the differential equations in Exercises ...
 9.3.12: Find all real solutions of the differential equations in Exercises ...
 9.3.13: Find all real solutions of the differential equations in Exercises ...
 9.3.14: Find all real solutions of the differential equations in Exercises ...
 9.3.15: Find all real solutions of the differential equations in Exercises ...
 9.3.16: Find all real solutions of the differential equations in Exercises ...
 9.3.17: Find all real solutions of the differential equations in Exercises ...
 9.3.18: Find all real solutions of the differential equations in Exercises ...
 9.3.19: Find all real solutions of the differential equations in Exercises ...
 9.3.20: Find all real solutions of the differential equations in Exercises ...
 9.3.21: Find all real solutions of the differential equations in Exercises ...
 9.3.22: Find all real solutions of the differential equations in Exercises ...
 9.3.23: So/ve the initial value problems in Exercises 23 through 29.f'(t) ...
 9.3.24: So/ve the initial value problems in Exercises 23 through 29.^ + 3.v...
 9.3.25: So/ve the initial value problems in Exercises 23 through 29. /'(/) ...
 9.3.26: So/ve the initial value problems in Exercises 23 through 29./"(/) ...
 9.3.27: So/ve the initial value problems in Exercises 23 through 29./ " ( /...
 9.3.28: So/ve the initial value problems in Exercises 23 through 29./"(/) +...
 9.3.29: So/ve the initial value problems in Exercises 23 through 29./"(/) +...
 9.3.30: The temperature of a hot cup of coffee can be modeled by the DET'(t...
 9.3.31: The speed v(t) of a falling object can sometimes be modeled by dv m...
 9.3.32: Consider the balance B{t) of a bank account, with initial balance B...
 9.3.33: Consider a pendulum of length L. Let x(t) be the angle the pendulum...
 9.3.34: Consider a wooden block in the shape of a cube whose edges are 10 c...
 9.3.35: The displacement x it) of a certain oscillator can be modeled by th...
 9.3.36: The displacement jc (t) of a certain oscillator can be modeled by t...
 9.3.37: The displacement jc (/) of a certain oscillator can be modeled by t...
 9.3.38: a. If p(t) is a polynomial and k a scalar, show that(D  X)(p(t)eu ...
 9.3.39: Find all solutions of the linear DEfit) + 3/"(r) + 3/'(/) + fit) = ...
 9.3.40: Find all solutions of the linear DE</3jc d2 x dx + 7  ' = ' {Hint:...
 9.3.41: If T is an mhorder linear differential operator and k is an arbitr...
 9.3.42: Let be the space of all realvalued smooth functions. a. Consider t...
 9.3.43: The displacement of a certain forced oscillator can be modeled by t...
 9.3.44: The displacement of a certain forced oscillator can be modeled by t...
 9.3.45: Use Theorem 9.3.13 to solve the initial value problemdx dtJ, with J...
 9.3.46: Use Theorem 9.3.13 to solve the initial value problemdx I t[Hint: F...
 9.3.47: Consider the initial value problem dx _ = Ax, with jc(0) = xo, dt w...
 9.3.48: Consider an n x n matrix A with m distinct eigenvalues ^1 * km. a. ...
Solutions for Chapter 9.3: Linear Differential Operators and Linear Differential Equations
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 9.3: Linear Differential Operators and Linear Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.3: Linear Differential Operators and Linear Differential Equations includes 48 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Since 48 problems in chapter 9.3: Linear Differential Operators and Linear Differential Equations have been answered, more than 14915 students have viewed full stepbystep solutions from this chapter.

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).

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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

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.