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# Solutions for Chapter 10.5: Separation of Variables; Heat Conduction in a Rod

## Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition

ISBN: 9781119256007

Solutions for Chapter 10.5: Separation of Variables; Heat Conduction in a Rod

Solutions for Chapter 10.5
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##### ISBN: 9781119256007

This expansive textbook survival guide covers the following chapters and their solutions. Since 27 problems in chapter 10.5: Separation of Variables; Heat Conduction in a Rod have been answered, more than 12392 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Chapter 10.5: Separation of Variables; Heat Conduction in a Rod includes 27 full step-by-step solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Augmented matrix [A b].

Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Column space C (A) =

space of all combinations of the columns of A.

• Condition number

cond(A) = c(A) = IIAIlIIA-III = 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.

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

• Fibonacci numbers

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

• 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.

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• 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.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Spanning set.

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

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

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