 3.1.1: In each of 1 through 6, find the general solution of the given diff...
 3.1.2: In each of 1 through 6, find the general solution of the given diff...
 3.1.3: In each of 1 through 6, find the general solution of the given diff...
 3.1.4: In each of 1 through 6, find the general solution of the given diff...
 3.1.5: In each of 1 through 6, find the general solution of the given diff...
 3.1.6: In each of 1 through 6, find the general solution of the given diff...
 3.1.7: In each of 7 through 12, find the solution of the given initial val...
 3.1.8: In each of 7 through 12, find the solution of the given initial val...
 3.1.9: In each of 7 through 12, find the solution of the given initial val...
 3.1.10: In each of 7 through 12, find the solution of the given initial val...
 3.1.11: In each of 7 through 12, find the solution of the given initial val...
 3.1.12: In each of 7 through 12, find the solution of the given initial val...
 3.1.13: Find a differential equation whose general solution is y = c1e2t + ...
 3.1.14: Find the solution of the initial value problem y__ y = 0, y(0) = 5 ...
 3.1.15: Find the solution of the initial value problem 2y__ 3y_ + y = 0, y(...
 3.1.16: Solve the initial value problem y__ y_ 2y = 0, y(0) = , y_(0) = 2. ...
 3.1.17: In each of 17 and 18, determine the values of , if any, for which a...
 3.1.18: In each of 17 and 18, determine the values of , if any, for which a...
 3.1.19: Consider the initial value problem (see Example 5) y__ + 5y_ + 6y =...
 3.1.20: Consider the equation ay__ +by_ +cy = d, where a, b, c, and d are c...
 3.1.21: Consider the equation ay__ + by_ + cy = 0, where a, b, and c are co...
Solutions for Chapter 3.1: Homogeneous Differential Equations with Constant Coefficients
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 3.1: Homogeneous Differential Equations with Constant Coefficients
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.1: Homogeneous Differential Equations with Constant Coefficients includes 21 full stepbystep solutions. Since 21 problems in chapter 3.1: Homogeneous Differential Equations with Constant Coefficients have been answered, more than 12629 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11.

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

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)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.