 1.5.1.1.103: (CAUTION!) Show that e1n x = l/x (not x) and e1n(sec x) = cos x.
 1.5.1.1.104: (Integration constant) Give a reason why in (4) you may choose the ...
 1.5.1.1.105: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.106: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.107: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.108: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.109: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.110: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.111: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.112: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.113: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.114: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.115: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.116: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.117: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.118: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.119: Find the general solution. If an initial condition is given, find a...
 1.5.1.1.120: Using a method of this section or separating variables, find the ge...
 1.5.1.1.121: Using a method of this section or separating variables, find the ge...
 1.5.1.1.122: Using a method of this section or separating variables, find the ge...
 1.5.1.1.123: Using a method of this section or separating variables, find the ge...
 1.5.1.1.124: Using a method of this section or separating variables, find the ge...
 1.5.1.1.125: Using a method of this section or separating variables, find the ge...
 1.5.1.1.126: Using a method of this section or separating variables, find the ge...
 1.5.1.1.127: (Investment programs) Bill opens a retirement savings account with ...
 1.5.1.1.128: (Mixing problem) A tank (as in Fig. 9 in Sec. 1.3) contains 1000 ga...
 1.5.1.1.129: (Lake Erie) Lake Erie has a water volume of about 450 km3 and a flo...
 1.5.1.1.130: (Heating and cooling of a building) Heating and cooling of a buildi...
 1.5.1.1.131: (Drug injection) Find and solve the model for drug injection into t...
 1.5.1.1.132: (Epidemics) A model for the spread of contagious diseases is obtain...
 1.5.1.1.133: (Extinction vs. unlimited growth) If in a population y(t) the death...
 1.5.1.1.134: (Harvesting renewable resources. Fishing) Suppose that the populati...
 1.5.1.1.135: (Harvesting) In Prob. 32 find and graph the solution satisfying yeO...
 1.5.1.1.136: (Intermittent harvesting) In Prob. 32 assume that you fish for 3 ye...
 1.5.1.1.137: (Harvesting) If a population of mice (in multiples of 1000) follows...
 1.5.1.1.138: (Harvesting) Do you save work in Prob. 34 if you first transform th...
 1.5.1.1.139: The sum YI + Y2 of two solutions YI and Y2 of the homogeneous equat...
 1.5.1.1.140: Y = 0 (that is, .v(x) = 0 for all x, also written y(x) "'" 0) is a ...
 1.5.1.1.141: The sum of a solution of (I) and a solution of (2) is a solution of...
 1.5.1.1.142: The difference of two solutions of (l) is a solution of (2).
 1.5.1.1.143: If Yl is a sulution of (I), what can you say about eYl?
 1.5.1.1.144: If YI and Y2 are solutions of y~ + PYI = rl and Y~ + PY2 = r2, resp...
 1.5.1.1.145: CAS EXPERIMENT. (a) Solve the ODE y'  ylx = x1 cos (l/x). Find a...
 1.5.1.1.146: TEAM PROJECT. Riccati Equation, Clairaut Equation. A Riccati equati...
 1.5.1.1.147: (Variation of parameter) Another method of obtaining (4) results fr...
 1.5.1.1.148: TEAM PROJECT. Transformations of ODEs. We have transformed ODEs to ...
Solutions for Chapter 1.5: Linear ODEs. Bernoulli Equation. Population Dynamics
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 1.5: Linear ODEs. Bernoulli Equation. Population Dynamics
Get Full SolutionsChapter 1.5: Linear ODEs. Bernoulli Equation. Population Dynamics includes 46 full stepbystep solutions. Since 46 problems in chapter 1.5: Linear ODEs. Bernoulli Equation. Population Dynamics have been answered, more than 49075 students have viewed full stepbystep solutions from this chapter. 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. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

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.

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).

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.

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!

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

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