 4.4.1.436: The only solution of the initialvalue problem y x2y 0, y(0) 0, y(0...
 4.4.1.437: For the method of undetermined coefficients, the assumed form of th...
 4.4.1.438: A constant multiple of a solution of a linear differential equation...
 4.4.1.439: If the set consisting of two functions f1 and f2 is linearly indepe...
 4.4.1.440: If is a solution of a homogeneous linear secondorder differential ...
 4.4.1.441: If is a solution of a homogeneous fourthorder linear differential ...
 4.4.1.442: If is the general solution of a homogeneous secondorder CauchyEul...
 4.4.1.443: is particular solution of for
 4.4.1.444: If is a particular solution of and is a particular solution of then...
 4.4.1.445: If and are solutions of homogeneous linear differential equation, t...
 4.4.1.446: Give an interval over which the set of two functions f1(x) x2 and f...
 4.4.1.447: Without the aid of the Wronskian, determine whether the given set o...
 4.4.1.448: Suppose m1 3, m2 5, and m3 1 are roots of multiplicity one, two, an...
 4.4.1.449: Consider the differential equation ay by cy g(x), where a, b, and c...
 4.4.1.450: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.451: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.452: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.453: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.454: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.455: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.456: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.457: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.458: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.459: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.460: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.461: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.462: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.463: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.464: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.465: In 1530 use the procedures developed in this chapter to find the ge...
 4.4.1.466: Write down the form of the general solution y yc yp of the given di...
 4.4.1.467: (a) Given that y sin x is a solution of y(4) 2y 11y 2y 10y 0, find ...
 4.4.1.468: (a) Write the general solution of the fourthorder DE y(4) 2y y 0 e...
 4.4.1.469: Consider the differential equation x2y (x2 2x)y (x 2)y x3 . Verify ...
 4.4.1.470: In 3540 solve the given differential equation subject to the indica...
 4.4.1.471: In 3540 solve the given differential equation subject to the indica...
 4.4.1.472: In 3540 solve the given differential equation subject to the indica...
 4.4.1.473: In 3540 solve the given differential equation subject to the indica...
 4.4.1.474: In 3540 solve the given differential equation subject to the indica...
 4.4.1.475: In 3540 solve the given differential equation subject to the indica...
 4.4.1.476: (a) Use a CAS as an aid in finding the roots of the auxiliary equat...
 4.4.1.477: Find a member of the family of solutions of whose graph is tangent ...
 4.4.1.478: In 4346 use systematic elimination to solve the given system.dx dt ...
 4.4.1.479: In 4346 use systematic elimination to solve the given system. dxdt ...
 4.4.1.480: In 4346 use systematic elimination to solve the given system.(D 2)x...
 4.4.1.481: In 4346 use systematic elimination to solve the given system. (D 2 ...
Solutions for Chapter 4: HigherOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 4: HigherOrder Differential Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 4: HigherOrder Differential Equations includes 46 full stepbystep solutions. Since 46 problems in chapter 4: HigherOrder Differential Equations have been answered, more than 20425 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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.

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.

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.

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

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

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

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