 1.9.1: Use the theorem of equality of mixed partial derivatives to show th...
 1.9.2: Show that the expression M(t,y) j(il~(r,~)/ilt)dj. is a function o...
 1.9.3: In each of 36 find the general solution of the given differential ...
 1.9.4: In each of 36 find the general solution of the given differential ...
 1.9.5: In each of 36 find the general solution of the given differential ...
 1.9.6: In each of 36 find the general solution of the given differential ...
 1.9.7: In each of 71 1, solve the given initialvalue problem.
 1.9.8: In each of 71 1, solve the given initialvalue problem.
 1.9.9: In each of 71 1, solve the given initialvalue problem.
 1.9.10: In each of 71 1, solve the given initialvalue problem.
 1.9.11: In each of 71 1, solve the given initialvalue problem.
 1.9.12: In each of 1214, determine the constant a so that the equation is ...
 1.9.13: In each of 1214, determine the constant a so that the equation is ...
 1.9.14: In each of 1214, determine the constant a so that the equation is ...
 1.9.15: Show that every separable equation of the form M(t)+ N(y)dy/dt=O is...
 1.9.16: Find all functions f(t) suoh that the differential equation y2sint ...
 1.9.17: Show that if ((aN/ at)  (aM/ ay))/ M = Q (y), then the differentia...
 1.9.18: The differential equation f (t)(dy /dt) + t2 + y = 0 is known to ha...
 1.9.19: The differential equation et secy  tany + (&/dt) = 0 has an integr...
 1.9.20: The Bernoulli differential equation is (dy /dt) + a(t) y = b(t) y "...
Solutions for Chapter 1.9: Exact equations, and why we cannot solve very many differential equations
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 1.9: Exact equations, and why we cannot solve very many differential equations
Get Full SolutionsSince 20 problems in chapter 1.9: Exact equations, and why we cannot solve very many differential equations have been answered, more than 5912 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.9: Exact equations, and why we cannot solve very many differential equations includes 20 full stepbystep solutions. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069.

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

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

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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).

Column space C (A) =
space of all combinations of the columns of A.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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 •

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

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.

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