 3.3.3.1.42: What is the superposition or linearity principle? For what 11thord...
 3.3.3.1.43: List some other basic theorems that extend from secondorder to 11t...
 3.3.3.1.44: If you know a general solution of a homogeneous linear ODE. what do...
 3.3.3.1.45: What is an initial value problem for an 11thorder linear ODE?
 3.3.3.1.46: What is the Wronskian? What is it used for?
 3.3.3.1.47: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.48: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.49: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.50: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.51: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.52: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.53: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.54: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.55: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.56: Solve the given ODE. (Show the details of your work.)
 3.3.3.1.57: Solve the given problem. (Show the details.)
 3.3.3.1.58: Solve the given problem. (Show the details.)
 3.3.3.1.59: Solve the given problem. (Show the details.)
 3.3.3.1.60: Solve the given problem. (Show the details.)
 3.3.3.1.61: Solve the given problem. (Show the details.)
Solutions for Chapter 3.3: Nonhomogeneous Linear ODEs
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 3.3: Nonhomogeneous Linear ODEs
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Since 20 problems in chapter 3.3: Nonhomogeneous Linear ODEs have been answered, more than 46347 students have viewed full stepbystep solutions from this chapter. Chapter 3.3: Nonhomogeneous Linear ODEs includes 20 full stepbystep solutions.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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].

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.

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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