 5.4.1: In each of 1 through 12 determine the general solution of the given...
 5.4.2: In each of 1 through 12 determine the general solution of the given...
 5.4.3: In each of 1 through 12 determine the general solution of the given...
 5.4.4: In each of 1 through 12 determine the general solution of the given...
 5.4.5: In each of 1 through 12 determine the general solution of the given...
 5.4.6: In each of 1 through 12 determine the general solution of the given...
 5.4.7: In each of 1 through 12 determine the general solution of the given...
 5.4.8: In each of 1 through 12 determine the general solution of the given...
 5.4.9: In each of 1 through 12 determine the general solution of the given...
 5.4.10: In each of 1 through 12 determine the general solution of the given...
 5.4.11: In each of 1 through 12 determine the general solution of the given...
 5.4.12: In each of 1 through 12 determine the general solution of the given...
 5.4.13: In each of 13 through 16 find the solution of the given initial val...
 5.4.14: In each of 13 through 16 find the solution of the given initial val...
 5.4.15: In each of 13 through 16 find the solution of the given initial val...
 5.4.16: In each of 13 through 16 find the solution of the given initial val...
 5.4.17: In each of 17 through 34 find all singular points of the given equa...
 5.4.18: In each of 17 through 34 find all singular points of the given equa...
 5.4.19: In each of 17 through 34 find all singular points of the given equa...
 5.4.20: In each of 17 through 34 find all singular points of the given equa...
 5.4.21: In each of 17 through 34 find all singular points of the given equa...
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 5.4.29: In each of 17 through 34 find all singular points of the given equa...
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 5.4.33: In each of 17 through 34 find all singular points of the given equa...
 5.4.34: In each of 17 through 34 find all singular points of the given equa...
 5.4.35: Find all values of for which all solutions of x2y + xy + (5/2)y = 0...
 5.4.36: Find all values of for which all solutions of x2y + y = 0 approach ...
 5.4.37: Find so that the solution of the initial value problem x2y 2y = 0, ...
 5.4.38: Find all values of for which all solutions of x2y + xy + (5/2)y = 0...
 5.4.39: Consider the Euler equation x2y + xy + y = 0. Find conditions on an...
 5.4.40: Using the method of reduction of order, show that if r1 is a repeat...
 5.4.41: In each of 41 and 42 show that the point x = 0 is a regular singula...
 5.4.42: In each of 41 and 42 show that the point x = 0 is a regular singula...
 5.4.43: Singularities at Infinity. The definitions of an ordinary point and...
 5.4.44: In each of 44 through 49 use the results of to determine whether th...
 5.4.45: In each of 44 through 49 use the results of to determine whether th...
 5.4.46: In each of 44 through 49 use the results of to determine whether th...
 5.4.47: In each of 44 through 49 use the results of to determine whether th...
 5.4.48: In each of 44 through 49 use the results of to determine whether th...
 5.4.49: In each of 44 through 49 use the results of to determine whether th...
Solutions for Chapter 5.4: Euler Equations; Regular Singular Points
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 5.4: Euler Equations; Regular Singular Points
Get Full SolutionsSince 49 problems in chapter 5.4: Euler Equations; Regular Singular Points have been answered, more than 13818 students have viewed full stepbystep solutions from this chapter. Chapter 5.4: Euler Equations; Regular Singular Points includes 49 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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!).

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

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

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

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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