- 1.1.1: dy dt = y + 3 1 y
- 1.1.2: dy dt = (t2 1)(y2 2) y2 4
- 1.1.3: Consider the population model d P dt = 0.4P 1 P 230 , where P(t) is...
- 1.1.4: Consider the population model d P dt = 0.3 1 P 200 P 50 1 P, where ...
- 1.1.5: Consider the differential equation dy dt = y3 y2 12y. (a) For what ...
- 1.1.6: Model radioactive decay using the notation t = time (independent va...
- 1.1.7: The half-life of a radioactive isotope is the amount of time it tak...
- 1.1.8: Carbon dating is a method of determining the time elapsed since the...
- 1.1.9: Engineers and scientists often measure the rate of decay of an expo...
- 1.1.10: The radioactive isotope I-131 is used in the treatment of hyperthyr...
- 1.1.11: MacQuarie Island is a small island about half-way between Antarctic...
- 1.1.12: The velocity v of a freefalling skydiver is well modeled by the dif...
- 1.1.13: For what value of L, 0 L 1, does learning occur most rapidly?
- 1.1.14: Suppose two students memorize lists according to the model d L dt =...
- 1.1.15: Consider the following two differential equations that model two st...
- 1.1.16: The expenditure on education in the U.S. is given in the following ...
- 1.1.17: Suppose a species of fish in a particular lake has a population tha...
- 1.1.18: Suppose that the growth-rate parameter k = 0.3 and the carrying cap...
- 1.1.19: The rhinoceros is now extremely rare. Suppose enough game preserve ...
- 1.1.20: While it is difficult to imagine a time before cell phones, such a ...
- 1.1.21: For the following predator-prey systems, identify which dependent v...
- 1.1.22: In the following predator-prey population models, x represents the ...
- 1.1.23: The following systems are models of the populations of pairs of spe...
Solutions for Chapter 1.1: MODELING VIA DIFFERENTIAL EQUATIONS
Full solutions for Differential Equations 00 | 4th Edition
Tv = Av + Vo = linear transformation plus shift.
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.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
Gram-Schmidt 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.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
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).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.