 5.2.1: In 110, determine the singular points of the given differential equ...
 5.2.2: In 110, determine the singular points of the given differential equ...
 5.2.3: In 110, determine the singular points of the given differential equ...
 5.2.4: In 110, determine the singular points of the given differential equ...
 5.2.5: In 110, determine the singular points of the given differential equ...
 5.2.6: In 110, determine the singular points of the given differential equ...
 5.2.7: In 110, determine the singular points of the given differential equ...
 5.2.8: In 110, determine the singular points of the given differential equ...
 5.2.9: In 110, determine the singular points of the given differential equ...
 5.2.10: In 110, determine the singular points of the given differential equ...
 5.2.11: In 11 and 12, put the given differential equation into the form (3)...
 5.2.12: In 11 and 12, put the given differential equation into the form (3)...
 5.2.13: In 13 and 14, x 0 is a regular singular point of the given differen...
 5.2.14: In 13 and 14, x 0 is a regular singular point of the given differen...
 5.2.15: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.16: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.17: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.18: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.19: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.20: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.21: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.22: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.23: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.24: In 1524, x 0 is a regular singular point of the given differential ...
 5.2.25: In 2530, x 0 is a regular singular point of the given differential ...
 5.2.26: In 2530, x 0 is a regular singular point of the given differential ...
 5.2.27: In 2530, x 0 is a regular singular point of the given differential ...
 5.2.28: In 2530, x 0 is a regular singular point of the given differential ...
 5.2.29: In 2530, x 0 is a regular singular point of the given differential ...
 5.2.30: In 2530, x 0 is a regular singular point of the given differential ...
 5.2.31: In 31 and 32, x 0 is a regular singular point of the given differen...
 5.2.32: In 31 and 32, x 0 is a regular singular point of the given differen...
 5.2.33: (a) The differential equation x4 y ly 0 has an irregular singular p...
 5.2.34: Buckling of a Tapered Column In Example 4 of Section 3.9, we saw th...
 5.2.35: Discuss how you would define a regular singular point for the linea...
 5.2.36: Each of the differential equations x3 y y 0 and x 2 y (3x 1)y y 0 h...
 5.2.37: We have seen that x 0 is a regular singular point of any CauchyEule...
Solutions for Chapter 5.2: Solutions about Singular Points
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 5.2: Solutions about Singular Points
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Chapter 5.2: Solutions about Singular Points includes 37 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Since 37 problems in chapter 5.2: Solutions about Singular Points have been answered, more than 34784 students have viewed full stepbystep solutions from this chapter.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } 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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).