 2.7.1: In each of 1 through 4:(a) Find approximate values of the solution ...
 2.7.2: In each of 1 through 4:(a) Find approximate values of the solution ...
 2.7.3: In each of 1 through 4:(a) Find approximate values of the solution ...
 2.7.4: In each of 1 through 4:(a) Find approximate values of the solution ...
 2.7.5: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.6: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.7: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.8: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.9: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.10: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.11: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.12: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.13: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.14: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.15: Consider the initial value problemy = 3t2/(3y2 4), y(1) = 0.(a) Use...
 2.7.16: Consider the initial value problemy = t2 + y2, y(0) = 1.Use Eulers ...
 2.7.17: Consider the initial value problemy = (y2 + 2ty)/(3 + t2), y(1) = 2...
 2.7.18: Consider the initial value problemy = ty + 0.1y3, y(0) = ,where is ...
 2.7.19: Consider the initial value problemy = y2 t2, y(0) = ,where is a giv...
 2.7.20: Convergence of Eulers Method. It can be shown that under suitable c...
 2.7.21: In each of 21 through 23, use the technique discussed in to show th...
 2.7.22: In each of 21 through 23, use the technique discussed in to show th...
 2.7.23: In each of 21 through 23, use the technique discussed in to show th...
Solutions for Chapter 2.7: Numerical Approximations: Eulers Method
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 2.7: Numerical Approximations: Eulers Method
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 23 problems in chapter 2.7: Numerical Approximations: Eulers Method have been answered, more than 28333 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Chapter 2.7: Numerical Approximations: Eulers Method includes 23 full stepbystep solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327.

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.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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].

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.