 2.1: Find one solution of the system dx/dt = xsin y and dy/dt = y co...
 2.2: Find all equilibrium points of the system dx/dt = y and dy/dt = ey ...
 2.3: Convert the secondorder differential equation d2 y/dt2 = 1 to a fi...
 2.4: Find the general solution of the system of equations in Exercise 3.
 2.5: Find all equilibrium points of the system dx/dt = y and dy/dt = sin...
 2.6: (x2 4)(y2 9) have? What are they?
 2.7: Is the function (x(t), y(t)) = (e6t , 2e3t ) a solution to the syst...
 2.8: Write the secondorder equation and the corresponding firstorder s...
 2.9: Find the general solution of the system dx/dt = 2x and dy/dt = 3y.
 2.10: Sketch the x(t) and y(t)graphs corresponding to the solution of t...
 2.11: Give an example of a firstorder system of differential equations w...
 2.12: Suppose that F(2, 1) = (3, 2). What is the result of one step of Eu...
 2.13: Sketch the solution curve for the initialvalue problem dx/dt = x, ...
 2.14: Suppose that all solutions of the system dx/dt = f (x, y) and dy/dt...
 2.15: The function (x(t), y(t)) = (e6t , 2e3t ) is a solution to the syst...
 2.16: The function x(t) = 2 for all t is an equilibrium solution of the s...
 2.17: Two different firstorder autonomous systems can have the same vect...
 2.18: Two different firstorder autonomous systems can have the same dire...
 2.19: The function (x(t), y(t)) = (sin t,sin t) is a solution of some fir...
 2.20: If the function (x1(t), y1(t)) = (cost,sin t) is a solution to an a...
 2.21: If the function (x1(t), y1(t)) = (cost,sin t) is a solution of a fi...
 2.22: MacQuarie Island is a small island about halfway between Antarctic...
 2.23: The solution curve corresponding to the initial condition (1, 0) in...
 2.24: The x(t) and y(t)graphs of the solution with (x(0), y(0)) = (1/2,...
 2.25: The solution with initial condition (x(0), y(0)) = (0, 1) is the sa...
 2.26: The function y(t) for the solution with initial condition (x(0), y(...
 2.27: The functions x(t) and y(t) for the solution with initial condition...
 2.28: The x(t) and y(t)graphs of the solution with initial condition (x...
 2.29: Consider the system dx dt = cos 2y dy dt = 2y x. (a) Find its equil...
 2.30: Consider a decoupled system of the form dx dt = f (x) dy dt = g(y)....
 2.31: In Exercises 3134, a solution curve in the x yplane and an initial...
 2.32: In Exercises 3134, a solution curve in the x yplane and an initial...
 2.33: In Exercises 3134, a solution curve in the x yplane and an initial...
 2.34: In Exercises 3134, a solution curve in the x yplane and an initial...
 2.35: Consider the partially decoupled system dx dt = x + 2y + 1 dy dt = ...
 2.36: Consider the partially decoupled system dx dt = x y dy dt = y + 1. ...
 2.37: A simple model of a glider flying along up and down but not left or...
Solutions for Chapter 2: FirstOrder Systems
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 2: FirstOrder Systems
Get Full SolutionsDifferential Equations 00 was written by and is associated to the ISBN: 9780495561989. Chapter 2: FirstOrder Systems includes 37 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. Since 37 problems in chapter 2: FirstOrder Systems have been answered, more than 17253 students have viewed full stepbystep solutions from this chapter.

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.)

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.

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

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).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.