 6.2.1: Find the general solution of each of the following systems: (a) y_ ...
 6.2.2: Solve each of the following initial value problems: (a) y_ 1 = y1 +...
 6.2.3: Given Y = c1e1tx1 + c2e2tx2 + +cnen txn is the solution to the init...
 6.2.4: Two tanks each contain 100 liters of a mixture. Initially, the mixt...
 6.2.5: Find the general solution of each of the following systems: (a) y__...
 6.2.6: Solve the initial value problem y__ 1 = 2y2 + y_ 1 + 2y_ 2 y__ 2 = ...
 6.2.7: In Application 2, assume that the solutions are of the form x1 = a1...
 6.2.8: Solve the the problem in Application 2, using the initial condition...
 6.2.9: Two masses are connected by springs as shown in the accompanying di...
 6.2.10: Three masses are connected by a series of springs between two fixed...
 6.2.11: Transform the nthorder equation y(n) = a0 y + a1 y_ + +an1 y(n1) i...
Solutions for Chapter 6.2: Systems of Linear Differential Equations
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 6.2: Systems of Linear Differential Equations
Get Full SolutionsSince 11 problems in chapter 6.2: Systems of Linear Differential Equations have been answered, more than 4186 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Chapter 6.2: Systems of Linear Differential Equations includes 11 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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

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

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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.

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