 2.1.1: Let L[y](t) = yM(t)  3ry'(t) + 3y(t). Compute (a) L[et1, (b) L[cos...
 2.1.2: Let L[y](t) = yl'(t)  6yf(t) + 5y(t). Compute (a) L[efl, (b) L[e2'...
 2.1.3: Show that the operator L defined by is linear; that is, L[cy]= cL[y...
 2.1.4: Let L[y](t) = yl'(t) +p(t) y'(t) + q(t) y(t), and suppose that ~[t~...
 2.1.5: (a) Show that y,(t) = fi and y2(t) = I /t are solutions of the diff...
 2.1.6: a) Show that yl(t)  e "/' and y2(t) = ef2/2/ies2/2u!s arc soluti...
 2.1.7: Compute the Wronskian of the following pairs of functions. (a) sin ...
 2.1.8: Let y,(t) and y2(t) be solutions of (3) on the interval  oo < t < ...
 2.1.9: (a) Let y ,(t) and y2(t) be solutions of (3) on the interval a < t ...
 2.1.10: Show that y(t)= t2 can never be a solution of (3) if the functions ...
 2.1.11: ~et~,(t)=t~ andy2(t)=tltl. (a) Show that yl and y2 are linearly dep...
 2.1.12: Suppose that y, andy2 are linearly independent on an interval I. Pr...
 2.1.13: Let y , and y2 be solutions of Bessel's equation on the interval 0 ...
 2.1.14: Suppose that the Wronskian of any two solutions of (3) is constant ...
 2.1.15: In 1518, assume that p and q are continuous, and that the function...
 2.1.16: In 1518, assume that p and q are continuous, and that the function...
 2.1.17: In 1518, assume that p and q are continuous, and that the function...
 2.1.18: In 1518, assume that p and q are continuous, and that the function...
 2.1.19: Suppose that W [y ,,y2J(t*) = 0, and, in addition, y ,(t*) = 0. Pro...
Solutions for Chapter 2.1: Algebraic properties of solutions
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 2.1: Algebraic properties of solutions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Chapter 2.1: Algebraic properties of solutions includes 19 full stepbystep solutions. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. Since 19 problems in chapter 2.1: Algebraic properties of solutions have been answered, more than 6534 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.

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

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.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.