 1.6.1: For 115, solve the given differential equation.dydx + y = 4ex .
 1.6.2: For 115, solve the given differential equation.
 1.6.3: For 115, solve the given differential equation.x2 y 4xy = x7 sin x,...
 1.6.4: For 115, solve the given differential equation.
 1.6.5: For 115, solve the given differential equation.dydx +2x1 x2 y = 4x,...
 1.6.6: For 115, solve the given differential equation.dydx +2x1 + x2 y = 4...
 1.6.7: For 115, solve the given differential equation.
 1.6.8: For 115, solve the given differential equation.
 1.6.9: For 115, solve the given differential equation.y y tan x = 8 sin3 x.
 1.6.10: For 115, solve the given differential equation.
 1.6.11: For 115, solve the given differential equation.
 1.6.12: For 115, solve the given differential equation.(1 y sin x)dx (cos x...
 1.6.13: For 115, solve the given differential equation.
 1.6.14: For 115, solve the given differential equation.y + y = ex , where ,...
 1.6.15: For 115, solve the given differential equation.y + mx1 y = ln x, wh...
 1.6.16: In 1621, solve the given initialvalue problem.y + 2x1 y = 4x, y(1)...
 1.6.17: In 1621, solve the given initialvalue problem.
 1.6.18: In 1621, solve the given initialvalue problem.dxdt +24 tx = 5, x(0...
 1.6.19: In 1621, solve the given initialvalue problem.
 1.6.20: In 1621, solve the given initialvalue problem.y + y = f (x), y(0) ...
 1.6.21: In 1621, solve the given initialvalue problem. 2y = f (x), y(0) = ...
 1.6.22: Solve the initialvalue problem in Example 1.6.5 as follows. First ...
 1.6.23: Find the general solution to the secondorder differential equation...
 1.6.24: Solve the differential equation for Newtons law of cooling by viewi...
 1.6.25: Suppose that an object is placed in a medium whose temperature is i...
 1.6.26: Between 8 a.m. and 12 p.m. on a hot summer day, the temperature ros...
 1.6.27: It is known that a certain object has constant of proportionality k...
 1.6.28: The differential equation dT dt = k1[T Tm(t)] + A0, (1.6.13) where ...
 1.6.29: This problem demonstrates the variationofparameters method for fir...
 1.6.30: For 3033, use the technique derived in the previous problem to solv...
 1.6.31: For 3033, use the technique derived in the previous problem to solv...
 1.6.32: For 3033, use the technique derived in the previous problem to solv...
 1.6.33: For 3033, use the technique derived in the previous problem to solv...
 1.6.34: For 3439, use a differential equations solver to determine the solu...
 1.6.35: For 3439, use a differential equations solver to determine the solu...
 1.6.36: For 3439, use a differential equations solver to determine the solu...
 1.6.37: For 3439, use a differential equations solver to determine the solu...
 1.6.38: For 3439, use a differential equations solver to determine the solu...
 1.6.39: For 3439, use a differential equations solver to determine the solu...
Solutions for Chapter 1.6: FirstOrder Linear Differential Equations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 1.6: FirstOrder Linear Differential Equations
Get Full SolutionsChapter 1.6: FirstOrder Linear Differential Equations includes 39 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. Since 39 problems in chapter 1.6: FirstOrder Linear Differential Equations have been answered, more than 19281 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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)ยท(b  Ax) = o.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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.

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.