 1.2.1: Solve each of the following initial value problems and plot the sol...
 1.2.2: Follow the instructions for for the following initialvalue problem...
 1.2.3: Consider the differential equation dy/dt = ay + b, where both a and...
 1.2.4: Consider the differential equation dy/dt = ay b. a. Find the equili...
 1.2.5: Undetermined Coefficients. Here is an alternative way to solve the ...
 1.2.6: Use the method of to solve the equation dy dt = ay + b.
 1.2.7: The field mouse population in Example 1 satisfies the differential ...
 1.2.8: The falling object in Example 2 satisfies the initial value problem...
 1.2.9: Consider the falling object of mass 10 kg in Example 2, but assume ...
 1.2.10: A radioactive material, such as the isotope thorium234, disintegra...
 1.2.11: The halflife of a radioactive material is the time required for an...
 1.2.12: According to Newtons law of cooling (see of Section 1.1), the tempe...
 1.2.13: Consider an electric circuit containing a capacitor, resistor, and ...
 1.2.14: A pond containing 1,000,000 gal of water is initially free of a cer...
Solutions for Chapter 1.2: Solutions of Some Differential Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 1.2: Solutions of Some Differential Equations
Get Full SolutionsSince 14 problems in chapter 1.2: Solutions of Some Differential Equations have been answered, more than 12661 students have viewed full stepbystep solutions from this chapter. Chapter 1.2: Solutions of Some Differential Equations includes 14 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11.

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.