 2.2.1: In 122, solve the given differential equation by separation of vari...
 2.2.2: In 122, solve the given differential equation by separation of vari...
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 2.2.23: In 2328, find an implicit and an explicit solution of the given ini...
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 2.2.29: In 29 and 30, proceed as in Example 5 and find an explicit solution...
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 2.2.31: In 3134, find an explicit solution of the given initialvalue proble...
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 2.2.35: (a) Find a solution of the initialvalue problem consisting of the ...
 2.2.36: Find a solution of x dy dx y2 2 y that passes through the indicated...
 2.2.37: Find a singular solution of 21. Of 22.
 2.2.38: Find a singular solution of 21. Of 22.
 2.2.39: Often a radical change in the form of the solution of a differentia...
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 2.2.43: Every autonomous firstorder equation dy/dx f (y) is separable. Fin...
 2.2.44: (a) The autonomous firstorder differential equation dy/dx 1/(y 3) ...
 2.2.45: (a) Find an explicit solution of the initialvalue problem dy dx 2x...
 2.2.46: Repeat parts (a)(c) of for the IVP consisting of the differential e...
 2.2.47: (a) Explain why the interval of definition of the explicit solution...
 2.2.48: On page 47 we showed that a oneparameter family of solutions of th...
 2.2.49: In 43 and 44 we saw that every autonomous firstorder differential e...
 2.2.50: Without the use of technology, how would you solve ("x x) dy dx "y ...
 2.2.51: Find a function whose square plus the square of its derivative is 1.
 2.2.52: (a) The differential equation in is equivalent to the normal form d...
 2.2.53: (a) The differential equation in is equivalent to the normal form d...
 2.2.54: (a) Use a CAS and the concept of level curves to plot representativ...
 2.2.55: (a) Find an implicit solution of the IVP (2y 2)dy 2 (4x3 6x)dx 0, y...
 2.2.56: (a) Use a CAS and the concept of level curves to plot representativ...
Solutions for Chapter 2.2: Separable Equations
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 2.2: Separable Equations
Get Full SolutionsSince 56 problems in chapter 2.2: Separable Equations have been answered, more than 36481 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2: Separable Equations includes 56 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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!

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 II· Condition numbers measure the sensitivity of the output to change in the input.

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

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].

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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).

Network.
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.

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.

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.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).