 3.4.1: In each of 1 through 10, find the general solution of the given dif...
 3.4.2: In each of 1 through 10, find the general solution of the given dif...
 3.4.3: In each of 1 through 10, find the general solution of the given dif...
 3.4.4: In each of 1 through 10, find the general solution of the given dif...
 3.4.5: In each of 1 through 10, find the general solution of the given dif...
 3.4.6: In each of 1 through 10, find the general solution of the given dif...
 3.4.7: In each of 1 through 10, find the general solution of the given dif...
 3.4.8: In each of 1 through 10, find the general solution of the given dif...
 3.4.9: In each of 1 through 10, find the general solution of the given dif...
 3.4.10: In each of 1 through 10, find the general solution of the given dif...
 3.4.11: In each of 11 through 14, solve the given initial value problem. Sk...
 3.4.12: In each of 11 through 14, solve the given initial value problem. Sk...
 3.4.13: In each of 11 through 14, solve the given initial value problem. Sk...
 3.4.14: In each of 11 through 14, solve the given initial value problem. Sk...
 3.4.15: Consider the initial value problem4y+ 12y+ 9y = 0, y(0) = 1, y(0) =...
 3.4.16: Consider the following modification of the initial value problem in...
 3.4.17: Consider the initial value problem4y+ 4y+ y = 0, y(0) = 1, y(0) = 2...
 3.4.18: Consider the initial value problem9y+ 12y+ 4y = 0, y(0) = a > 0, y(...
 3.4.19: Consider the equation ay+ by+ cy = 0. If the roots of the correspon...
 3.4.20: 20 through 22 indicate other ways of finding the second solution wh...
 3.4.21: 20 through 22 indicate other ways of finding the second solution wh...
 3.4.22: 20 through 22 indicate other ways of finding the second solution wh...
 3.4.23: In each of 23 through 30, use the method of reduction of order to f...
 3.4.24: In each of 23 through 30, use the method of reduction of order to f...
 3.4.25: In each of 23 through 30, use the method of reduction of order to f...
 3.4.26: In each of 23 through 30, use the method of reduction of order to f...
 3.4.27: In each of 23 through 30, use the method of reduction of order to f...
 3.4.28: In each of 23 through 30, use the method of reduction of order to f...
 3.4.29: In each of 23 through 30, use the method of reduction of order to f...
 3.4.30: In each of 23 through 30, use the method of reduction of order to f...
 3.4.31: The differential equationy+ (xy+ y) = 0arises in the study of the t...
 3.4.32: The method of can be extended to second order equations with variab...
 3.4.33: In each of 33 through 36, use the method of to find a second indepe...
 3.4.34: In each of 33 through 36, use the method of to find a second indepe...
 3.4.35: In each of 33 through 36, use the method of to find a second indepe...
 3.4.36: In each of 33 through 36, use the method of to find a second indepe...
 3.4.37: Behavior of Solutions as t . 37 through 39 are concerned with the b...
 3.4.38: Behavior of Solutions as t . 37 through 39 are concerned with the b...
 3.4.39: Behavior of Solutions as t . 37 through 39 are concerned with the b...
 3.4.40: Euler Equations. In each of 40 through 45, use the substitution int...
 3.4.41: Euler Equations. In each of 40 through 45, use the substitution int...
 3.4.42: Euler Equations. In each of 40 through 45, use the substitution int...
 3.4.43: Euler Equations. In each of 40 through 45, use the substitution int...
 3.4.44: Euler Equations. In each of 40 through 45, use the substitution int...
 3.4.45: Euler Equations. In each of 40 through 45, use the substitution int...
Solutions for Chapter 3.4: Repeated Roots; Reduction of Order
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 3.4: Repeated Roots; Reduction of Order
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Chapter 3.4: Repeated Roots; Reduction of Order includes 45 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 45 problems in chapter 3.4: Repeated Roots; Reduction of Order have been answered, more than 18055 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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

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

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Solvable system Ax = b.
The right side b is in the column space of A.

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.