 2.5.2.1.205: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.206: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.207: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.208: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.209: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.210: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.211: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.212: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.213: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.214: In 110 solve the given differential equation by using an appropriat...
 2.5.2.1.215: In 1114 solve the given initialvalue problem. xy2 dydx y3 x3, y(1) 2
 2.5.2.1.216: In 1114 solve the given initialvalue problem. (x2 2y2) dxdy xy, y(...
 2.5.2.1.217: In 1114 solve the given initialvalue problem. (x yey/x) dx xey/x d...
 2.5.2.1.218: In 1114 solve the given initialvalue problem. y dx x(ln x ln y 1) ...
 2.5.2.1.219: Each DE in 1522 is a Bernoulli equation. In 1520 solve the given di...
 2.5.2.1.220: Each DE in 1522 is a Bernoulli equation. In 1520 solve the given di...
 2.5.2.1.221: Each DE in 1522 is a Bernoulli equation. In 1520 solve the given di...
 2.5.2.1.222: Each DE in 1522 is a Bernoulli equation. In 1520 solve the given di...
 2.5.2.1.223: Each DE in 1522 is a Bernoulli equation. In 1520 solve the given di...
 2.5.2.1.224: Each DE in 1522 is a Bernoulli equation. In 1520 solve the given di...
 2.5.2.1.225: In 21 and 22 solve the given initialvalue problem. x2 dydx 2xy 3y4...
 2.5.2.1.226: In 21 and 22 solve the given initialvalue problem. y1/2 dydx y3/2 ...
 2.5.2.1.227: In 2328 solve the given differential equation by using an appropria...
 2.5.2.1.228: In 2328 solve the given differential equation by using an appropria...
 2.5.2.1.229: In 2328 solve the given differential equation by using an appropria...
 2.5.2.1.230: In 2328 solve the given differential equation by using an appropria...
 2.5.2.1.231: In 2328 solve the given differential equation by using an appropria...
 2.5.2.1.232: In 2328 solve the given differential equation by using an appropria...
 2.5.2.1.233: In 29 and 30 solve the given initialvalue problem. dydx cos(x y), ...
 2.5.2.1.234: In 29 and 30 solve the given initialvalue problem. dydx 3x 2y3x 2y...
 2.5.2.1.235: Explain why it is always possible to express any homogeneous differ...
 2.5.2.1.236: Put the homogeneous differential equation (5x2 2y2 ) dx xy dy 0 int...
 2.5.2.1.237: (a) Determine two singular solutions of the DE in 10. (b) If the in...
 2.5.2.1.238: In Example 3 the solution y(x) becomes unbounded as x : . Neverthel...
 2.5.2.1.239: The differential equation dydx P(x) Q(x)y R(x)y2 is known as Riccat...
 2.5.2.1.240: Determine an appropriate substitution to solve xy y ln(xy).
 2.5.2.1.241: Falling Chain In in Exercises 2.4 we saw that a mathematical model ...
 2.5.2.1.242: Population Growth In the study of population dynamics one of the mo...
Solutions for Chapter 2.5: FirstOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 2.5: FirstOrder Differential Equations
Get Full SolutionsChapter 2.5: FirstOrder Differential Equations includes 38 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 38 problems in chapter 2.5: FirstOrder Differential Equations have been answered, more than 21242 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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.

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].

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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 •

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).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.