 1.1.1.179: Explain the tenns ordinary d~fferellfial equatiol/ (ODE). partial d...
 1.1.1.180: What is an initial condition? How is this condition used in an init...
 1.1.1.181: What is a homogeneous linear ODE? A nonhomogeneous linear ODE? Why ...
 1.1.1.182: What do you know about direction fields and their practical importa...
 1.1.1.183: Give examples of mechanical problems that lead to ODEs.
 1.1.1.184: Why do electric circuits lead to ODEs?
 1.1.1.185: Make a list of the solution methods considered. Explain each method...
 1.1.1.186: Can certain ODEs be solved by more than one method? Give three exam...
 1.1.1.187: What are integrating factors? Explain the idea. Give examples.
 1.1.1.188: Does every firstorder ODE have a solution? A general solution? Wha...
 1.1.1.189: Graph a direction field (by a CAS or by hand) and sketch some of th...
 1.1.1.190: Graph a direction field (by a CAS or by hand) and sketch some of th...
 1.1.1.191: Graph a direction field (by a CAS or by hand) and sketch some of th...
 1.1.1.192: Graph a direction field (by a CAS or by hand) and sketch some of th...
 1.1.1.193: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.194: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.195: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.196: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.197: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.198: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.199: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.200: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.201: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.202: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.203: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.204: Find the general solution. Indicate which method in this chapter yo...
 1.1.1.205: Solve the following initial value problems. Indicate the method use...
 1.1.1.206: Solve the following initial value problems. Indicate the method use...
 1.1.1.207: Solve the following initial value problems. Indicate the method use...
 1.1.1.208: Solve the following initial value problems. Indicate the method use...
 1.1.1.209: Solve the following initial value problems. Indicate the method use...
 1.1.1.210: Solve the following initial value problems. Indicate the method use...
 1.1.1.211: ~Heat flow) If the isothelms in a region are x 2  )'2 = c, what ar...
 1.1.1.212: (Law of cooling) A thennometer showing WaC is brought into a room w...
 1.1.1.213: (Halflife) If 10o/c of a radioactive substance disintegrates in 4 ...
 1.1.1.214: (HaIflife) What is the halflife of a substance if after 5 days, 0...
 1.1.1.215: (HaIflife) When will 99% of the substance in Prob. 35 have disinte...
 1.1.1.216: (Air circulation) In a room containing 20000 ft3 of air, 600 ft3 of...
 1.1.1.217: (Electric field) If the equipotential lines in a region of the x)"...
 1.1.1.218: (Chemistry) In a bimolecular reaction A + B ? M, a moles per liter...
 1.1.1.219: (Population) Find the population y(1) if the birth rate is proporti...
 1.1.1.220: (Curves) Find all curve~ in the first quadrant of the Ayplane such ...
 1.1.1.221: (Optics) Lambert's law of absorption9 states that the absorption of...
Solutions for Chapter 1: FirstOrder ODEs
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 1: FirstOrder ODEs
Get Full SolutionsSince 43 problems in chapter 1: FirstOrder ODEs have been answered, more than 44128 students have viewed full stepbystep solutions from this chapter. Chapter 1: FirstOrder ODEs includes 43 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

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).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.