 5.4.1: In each of 1 through 12, determine the general solution of the give...
 5.4.2: In each of 1 through 12, determine the general solution of the give...
 5.4.3: In each of 1 through 12, determine the general solution of the give...
 5.4.4: In each of 1 through 12, determine the general solution of the give...
 5.4.5: In each of 1 through 12, determine the general solution of the give...
 5.4.6: In each of 1 through 12, determine the general solution of the give...
 5.4.7: In each of 1 through 12, determine the general solution of the give...
 5.4.8: In each of 1 through 12, determine the general solution of the give...
 5.4.9: In each of 1 through 12, determine the general solution of the give...
 5.4.10: In each of 1 through 12, determine the general solution of the give...
 5.4.11: In each of 1 through 12, determine the general solution of the give...
 5.4.12: In each of 1 through 12, determine the general solution of the give...
 5.4.13: In each of 13 through 16, find the solution of the given initial va...
 5.4.14: In each of 13 through 16, find the solution of the given initial va...
 5.4.15: In each of 13 through 16, find the solution of the given initial va...
 5.4.16: In each of 13 through 16, find the solution of the given initial va...
 5.4.17: In each of 17 through 34, find all singular points of the given equ...
 5.4.18: In each of 17 through 34, find all singular points of the given equ...
 5.4.19: In each of 17 through 34, find all singular points of the given equ...
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 5.4.28: In each of 17 through 34, find all singular points of the given equ...
 5.4.29: In each of 17 through 34, find all singular points of the given equ...
 5.4.30: In each of 17 through 34, find all singular points of the given equ...
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 5.4.33: In each of 17 through 34, find all singular points of the given equ...
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 5.4.35: Find all values of for which all solutions of x2y+ xy+ (5/2)y = 0 a...
 5.4.36: Find all values of for which all solutions of x2y+ y = 0 approach z...
 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 a...
 5.4.39: Consider the Euler equation x2y+ xy+ y = 0. Find conditions on and ...
 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 singul...
 5.4.42: In each of 41 and 42, show that the point x = 0 is a regular singul...
 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 t...
 5.4.45: In each of 44 through 49, use the results of to determine whether t...
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 5.4.47: In each of 44 through 49, use the results of to determine whether t...
 5.4.48: In each of 44 through 49, use the results of to determine whether t...
 5.4.49: In each of 44 through 49, use the results of to determine whether t...
Solutions for Chapter 5.4: Euler Equations; Regular Singular Points
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 5.4: Euler Equations; Regular Singular Points
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.4: Euler Equations; Regular Singular Points includes 49 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Since 49 problems in chapter 5.4: Euler Equations; Regular Singular Points have been answered, more than 16837 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

Column space C (A) =
space of all combinations of the columns of A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

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 ().

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

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.

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).