 1.8.1: dy dt = 4y + 9et
 1.8.2: dy dt = 4y + 3et
 1.8.3: dy dt = 3y + 4 cos 2t
 1.8.4: dy dt = 2y + sin 2t
 1.8.5: dy dt = 3y 4e3t
 1.8.6: dy dt = y 2 + 4et/2
 1.8.7: dy dt + 2y = et/3, y(0) = 1
 1.8.8: dy dt 2y = 3e2t , y(0) = 10
 1.8.9: dy dt + y = cos 2t, y(0) = 5
 1.8.10: dy dt + 3y = cos 2t, y(0) = 1
 1.8.11: dy dt 2y = 7e2t , y(0) = 3
 1.8.12: dy dt 2y = 7e2t , y(0) = 3
 1.8.13: Consider the nonhomogeneous linear equation dy dt + 2y = cos 3t. To...
 1.8.14: Consider the nonhomogeneous linear equation dy dt = y + cos 2t.To f...
 1.8.15: The graph to the right is the graph of a solution of a homogeneous ...
 1.8.16: The two graphs to the right are graphs of solutions of a nonhomogen...
 1.8.17: Consider the nonlinear differential equation dy/dt = y2. (a) Show t...
 1.8.18: Consider the nonhomogeneous linear equation dy/dt = y + 2. (a) Comp...
 1.8.19: Consider a nonhomogeneous linear equation of the form dy dt + a(t)y...
 1.8.20: Consider the nonhomogeneous linear equation dy dt + 2y = 3t 2 + 2t ...
 1.8.21: dy dt + 2y = t 2 + 2t + 1 + e4t
 1.8.22: dy dt + y = t 3 + sin 3t
 1.8.23: dy dt 3y = 2t e4t
 1.8.24: dy dt + y = cos 2t + 3 sin 2t + et
 1.8.25: dy dt + 2y = b(t), where 1 < b(t) < 2 for all t.
 1.8.26: dy dt 2y = b(t), where 1 < b(t) < 2 for all t.
 1.8.27: dy dt + y = b(t), where b(t) 3 as t .
 1.8.28: dy dt + ay = cos 3t + b, where a and b are positive constants.
 1.8.29: A person initially places $1,000 in a savings account that pays int...
 1.8.30: A student has saved $70,000 for her college tuition. When she start...
 1.8.31: A college professor contributes $5,000 per year into her retirement...
 1.8.32: Verify that the function y(t) = t/5 satisfies the nonhomogeneous li...
 1.8.33: In this exercise, we verify the Extended Linearity Principle for th...
 1.8.34: Suppose that every constant multiple of a solution is also a soluti...
Solutions for Chapter 1.8: LINEAR EQUATIONS
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 1.8: LINEAR EQUATIONS
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.8: LINEAR EQUATIONS includes 34 full stepbystep solutions. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. Since 34 problems in chapter 1.8: LINEAR EQUATIONS have been answered, more than 23330 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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