 6.4.1: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.2: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.3: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.4: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.5: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.6: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.7: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.8: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.9: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.10: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.11: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.12: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.13: In each of 1 through 13:(a) Find the solution of the given initial ...
 6.4.14: Find an expression involving uc(t) for a function f that ramps up f...
 6.4.15: Find an expression involving uc(t) for a function g that ramps up f...
 6.4.16: A certain springmass system satisfies the initial value problemu+ 1...
 6.4.17: Modify the problem in Example 2 of this section by replacing the gi...
 6.4.18: Consider the initial value problemy+ 13 y+ 4y = fk(t), y(0) = 0, y(...
 6.4.19: Consider the initial value problemy+ y = f(t), y(0) = 0, y(0) = 0,w...
 6.4.20: Consider the initial value problemy+ 0.1y+ y = f(t), y(0) = 0, y(0)...
 6.4.21: Consider the initial value problemy+ y = g(t), y(0) = 0, y(0) = 0,w...
 6.4.22: Consider the initial value problemy+ 0.1y+ y = g(t), y(0) = 0, y(0)...
 6.4.23: Consider the initial value problemy+ y = h(t), y(0) = 0, y(0) = 0,w...
Solutions for Chapter 6.4: Differential Equations with Discontinuous Forcing Functions
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 6.4: Differential Equations with Discontinuous Forcing Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Chapter 6.4: Differential Equations with Discontinuous Forcing Functions includes 23 full stepbystep solutions. Since 23 problems in chapter 6.4: Differential Equations with Discontinuous Forcing Functions have been answered, more than 16442 students have viewed full stepbystep solutions from this chapter. 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: 9780470458310.

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.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.