 2.1: In each of 1 through 24, solve the given differential equation. If ...
 2.2: In each of 1 through 24, solve the given differential equation. If ...
 2.3: In each of 1 through 24, solve the given differential equation. If ...
 2.4: In each of 1 through 24, solve the given differential equation. If ...
 2.5: In each of 1 through 24, solve the given differential equation. If ...
 2.6: In each of 1 through 24, solve the given differential equation. If ...
 2.7: In each of 1 through 24, solve the given differential equation. If ...
 2.8: In each of 1 through 24, solve the given differential equation. If ...
 2.9: In each of 1 through 24, solve the given differential equation. If ...
 2.10: In each of 1 through 24, solve the given differential equation. If ...
 2.11: In each of 1 through 24, solve the given differential equation. If ...
 2.12: In each of 1 through 24, solve the given differential equation. If ...
 2.13: In each of 1 through 24, solve the given differential equation. If ...
 2.14: In each of 1 through 24, solve the given differential equation. If ...
 2.15: In each of 1 through 24, solve the given differential equation. If ...
 2.16: In each of 1 through 24, solve the given differential equation. If ...
 2.17: In each of 1 through 24, solve the given differential equation. If ...
 2.18: In each of 1 through 24, solve the given differential equation. If ...
 2.19: In each of 1 through 24, solve the given differential equation. If ...
 2.20: In each of 1 through 24, solve the given differential equation. If ...
 2.21: In each of 1 through 24, solve the given differential equation. If ...
 2.22: In each of 1 through 24, solve the given differential equation. If ...
 2.23: In each of 1 through 24, solve the given differential equation. If ...
 2.24: In each of 1 through 24, solve the given differential equation. If ...
 2.25: Riccati Equations. The equation dy dt = q1(t) + q2(t) y + q3(t) y2 ...
 2.26: Verify that the given function is a particular solution of the give...
 2.27: The propagation of a single action in a large population (for examp...
 2.28: Equations with the Dependent Variable Missing. For a secondorder di...
 2.29: Equations with the Dependent Variable Missing. For a secondorder di...
 2.30: Equations with the Dependent Variable Missing. For a secondorder di...
 2.31: Equations with the Dependent Variable Missing. For a secondorder di...
 2.32: secondorder differential equations of the form y = f ( y, y), in w...
 2.33: secondorder differential equations of the form y = f ( y, y), in w...
 2.34: secondorder differential equations of the form y = f ( y, y), in w...
 2.35: secondorder differential equations of the form y = f ( y, y), in w...
 2.36: In each of 36 through 37, solve the given initial value problem usi...
 2.37: In each of 36 through 37, solve the given initial value problem usi...
Solutions for Chapter 2: FirstOrder Differential Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 2: FirstOrder Differential Equations
Get Full SolutionsSince 37 problems in chapter 2: FirstOrder Differential Equations have been answered, more than 13378 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2: FirstOrder Differential Equations includes 37 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007.

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.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Conjugate Gradient Method.
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.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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