 3.2.3.1.49: The number N(t) of supermarkets throughout the country that are usi...
 3.2.3.1.50: The number N(t) of people in a community who are exposed to a parti...
 3.2.3.1.51: A model for the population P(t) in a suburb of a large city is give...
 3.2.3.1.52: (a) Census data for the United States between 1790 and 1950 are giv...
 3.2.3.1.53: (a) If a constant number h of fish are harvested from a fishery per...
 3.2.3.1.54: Investigate the harvesting model in both qualitatively and analytic...
 3.2.3.1.55: Repeat in the case a 5, b 1, h 7
 3.2.3.1.56: (a) Suppose a b 1 in the Gompertz differential equation (7). Since ...
 3.2.3.1.57: Two chemicals A and B are combined to form a chemical C. The rate, ...
 3.2.3.1.58: Solve if 100 grams of chemical A is present initially. At what time...
 3.2.3.1.59: Leaking Cylindrical Tank A tank in the form of a rightcircular cyl...
 3.2.3.1.60: Leaking Cylindrical TankContinued When friction and contraction of ...
 3.2.3.1.61: Leaking Conical Tank A tank in the form of a rightcircular cone sta...
 3.2.3.1.62: Inverted Conical Tank Suppose that the conical tank in 13(a) is inv...
 3.2.3.1.63: Air Resistance A differential equation for the velocity v of a fall...
 3.2.3.1.64: How High?Nonlinear Air Resistance Consider the 16pound cannonball ...
 3.2.3.1.65: That Sinking Feeling (a) Determine a differential equation for the ...
 3.2.3.1.66: Solar Collector The differential equation describes the shape of a ...
 3.2.3.1.67: Tsunami (a) A simple model for the shape of a tsunami is given by ,...
 3.2.3.1.68: Evaporation An outdoor decorative pond in the shape of a hemispheri...
 3.2.3.1.69: Doomsday Equation Consider the differential equation where and In S...
 3.2.3.1.70: Doomsday or Extinction Suppose the population model (4) is modified...
 3.2.3.1.71: Regression Line Read the documentation for your CAS on scatter plot...
 3.2.3.1.72: Immigration Model (a) In Examples 3 and 4 of Section 2.1 we saw tha...
 3.2.3.1.73: What Goes Up... In let ta be the time it takes the cannonball to at...
 3.2.3.1.74: Skydiving A skydiver is equipped with a stopwatch and an altimeter....
 3.2.3.1.75: Hitting Bottom A helicopter hovers 500 feet above a large open tank...
 3.2.3.1.76: Old Man River . . . In Figure 3.2.8(a) suppose that the yaxis and ...
 3.2.3.1.77: (a) Solve the DE in subject to y(1) 0. For convenience let (b) Dete...
 3.2.3.1.78: Old Man River Keeps Moving... Suppose the man in again enters the c...
 3.2.3.1.79: The current speed vr of a straight river such as that in is usually...
 3.2.3.1.80: Raindrops Keep Falling... When a bottle of liquid refreshment was o...
 3.2.3.1.81: Time Drips By The clepsydra, or water clock, was a device that the ...
 3.2.3.1.82: (a) Suppose that a glass tank has the shape of a cone with circular...
 3.2.3.1.83: Suppose that r f(h) defines the shape of a water clock for which th...
Solutions for Chapter 3.2: Modeling with FirstOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 3.2: Modeling with FirstOrder Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 3.2: Modeling with FirstOrder Differential Equations includes 35 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This expansive textbook survival guide covers the following chapters and their solutions. Since 35 problems in chapter 3.2: Modeling with FirstOrder Differential Equations have been answered, more than 23185 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.