 1.2.1: In each of 17 find the general solution of the given differential ...
 1.2.2: In each of 17 find the general solution of the given differential ...
 1.2.3: In each of 17 find the general solution of the given differential ...
 1.2.4: In each of 17 find the general solution of the given differential ...
 1.2.5: In each of 17 find the general solution of the given differential ...
 1.2.6: In each of 17 find the general solution of the given differential ...
 1.2.7: In each of 17 find the general solution of the given differential ...
 1.2.8: In each of 814, find the solution of the given initialvalue problem.
 1.2.9: In each of 814, find the solution of the given initialvalue problem.
 1.2.10: In each of 814, find the solution of the given initialvalue problem.
 1.2.11: In each of 814, find the solution of the given initialvalue problem.
 1.2.12: In each of 814, find the solution of the given initialvalue problem.
 1.2.13: In each of 814, find the solution of the given initialvalue problem.
 1.2.14: In each of 814, find the solution of the given initialvalue problem.
 1.2.15: Find the general solution of the equation 4 2 5/2 (1+t2)+ty=(l+t )...
 1.2.16: Find the solution of the initialvalue problem
 1.2.17: Find a continuous solution of the initialvalue problem y'+y=g(O, y...
 1.2.18: Show that every solution of the equation (&/dt)+ay= be" where a an...
 1.2.19: Given the differential equation (&/dl)+ a(t)y = f (t) with a(t) and...
Solutions for Chapter 1.2: Firstorder linear differential equations
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 1.2: Firstorder linear differential equations
Get Full SolutionsChapter 1.2: Firstorder linear differential equations includes 19 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Since 19 problems in chapter 1.2: Firstorder linear differential equations have been answered, more than 5851 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

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

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