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# 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

Solutions for Chapter 5.4
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##### ISBN: 9780470383346

Since 49 problems in chapter 5.4: Euler Equations; Regular Singular Points have been answered, more than 13818 students have viewed full step-by-step solutions from this chapter. Chapter 5.4: Euler Equations; Regular Singular Points includes 49 full step-by-step 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.

Key Math Terms and definitions covered in this textbook
• 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!

• Cayley-Hamilton 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 S-I 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, off-diagonal 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 = Q-1 = 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.

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