 3.4.1: In 126, solve the given differential equation by undetermined coeff...
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 3.4.27: In 2736, solve the given initialvalue problem.
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 3.4.35: In 2736, solve the given initialvalue problem.
 3.4.36: In 2736, solve the given initialvalue problem.
 3.4.37: In 3740, solve the given boundaryvalue problem.
 3.4.38: In 3740, solve the given boundaryvalue problem.
 3.4.39: In 3740, solve the given boundaryvalue problem.
 3.4.40: In 3740, solve the given boundaryvalue problem.
 3.4.41: In 41 and 42, solve the given initialvalue problem in which the in...
 3.4.42: In 41 and 42, solve the given initialvalue problem in which the in...
 3.4.43: Consider the differential equation ay by cy ekx, where a, b, c, and...
 3.4.44: Discuss how the method of this section can be used to find a partic...
 3.4.45: In 4548, without solving match a solution curve of y y f (x) shown ...
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Solutions for Chapter 3.4: Undetermined Coefficients
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 3.4: Undetermined Coefficients
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This expansive textbook survival guide covers the following chapters and their solutions. Since 50 problems in chapter 3.4: Undetermined Coefficients have been answered, more than 32334 students have viewed full stepbystep solutions from this chapter. Chapter 3.4: Undetermined Coefficients includes 50 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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