 3.2.1: In each of 1 through 6, find the Wronskian of the given pair of fun...
 3.2.2: In each of 1 through 6, find the Wronskian of the given pair of fun...
 3.2.3: In each of 1 through 6, find the Wronskian of the given pair of fun...
 3.2.4: In each of 1 through 6, find the Wronskian of the given pair of fun...
 3.2.5: In each of 1 through 6, find the Wronskian of the given pair of fun...
 3.2.6: In each of 1 through 6, find the Wronskian of the given pair of fun...
 3.2.7: In each of 7 through 12, determine the longest interval in which th...
 3.2.8: In each of 7 through 12, determine the longest interval in which th...
 3.2.9: In each of 7 through 12, determine the longest interval in which th...
 3.2.10: In each of 7 through 12, determine the longest interval in which th...
 3.2.11: In each of 7 through 12, determine the longest interval in which th...
 3.2.12: In each of 7 through 12, determine the longest interval in which th...
 3.2.13: Verify that y1(t) = t2 and y2(t) = t1 are two solutions of the diff...
 3.2.14: Verify that y1(t) = 1 and y2(t) = t1/2 are solutions of the differe...
 3.2.15: . Show that if y = (t) is a solution of the differential equation y...
 3.2.16: Can y = sin(t2) be a solution on an interval containing t = 0 of an...
 3.2.17: If the Wronskian W of f and g is 3e4t, and if f(t) = e2t, find g(t).
 3.2.18: If the Wronskian W of f and g is t2et, and if f(t) = t, find g(t)
 3.2.19: If W(f, g) is the Wronskian of f and g, and if u = 2f g, v = f + 2g...
 3.2.20: If the Wronskian of f and g is t cost sin t, and if u = f + 3g, v =...
 3.2.21: Assume that y1 and y2 are a fundamental set of solutions of y + p(t...
 3.2.22: In each of 22 and 23, find the fundamental set of solutions specifi...
 3.2.23: In each of 22 and 23, find the fundamental set of solutions specifi...
 3.2.24: In each of 24 through 27, verify that the functions y1 and y2 are s...
 3.2.25: In each of 24 through 27, verify that the functions y1 and y2 are s...
 3.2.26: In each of 24 through 27, verify that the functions y1 and y2 are s...
 3.2.27: In each of 24 through 27, verify that the functions y1 and y2 are s...
 3.2.28: Consider the equation y y 2y = 0.(a) Show that y1(t) = et and y2(t)...
 3.2.29: In each of 29 through 32, find theWronskian of two solutions of the...
 3.2.30: In each of 29 through 32, find theWronskian of two solutions of the...
 3.2.31: In each of 29 through 32, find theWronskian of two solutions of the...
 3.2.32: In each of 29 through 32, find theWronskian of two solutions of the...
 3.2.33: Show that if p is differentiable and p(t) > 0, then the Wronskian W...
 3.2.34: If the differential equation ty + 2y + tety = 0 has y1 and y2 as a ...
 3.2.35: If the differential equation t2y 2y + (3 + t)y = 0 has y1 and y2 as...
 3.2.36: If theWronskian of any two solutions of y + p(t)y + q(t)y = 0 is co...
 3.2.37: If f, g, and h are differentiable functions, show that W(fg, fh) = ...
 3.2.38: In 38 through 40, assume that p and q are continuous and that the f...
 3.2.39: In 38 through 40, assume that p and q are continuous and that the f...
 3.2.40: In 38 through 40, assume that p and q are continuous and that the f...
 3.2.41: Exact Equations. The equationP(x)y + Q(x)y + R(x)y = 0is said to be...
 3.2.42: In each of 42 through 45, use the result of to determine whether th...
 3.2.43: In each of 42 through 45, use the result of to determine whether th...
 3.2.44: In each of 42 through 45, use the result of to determine whether th...
 3.2.45: In each of 42 through 45, use the result of to determine whether th...
 3.2.46: TheAdjoint Equation. If a second order linear homogeneous equation ...
 3.2.47: In each of 47 through 49, use the result of to find the adjoint of ...
 3.2.48: In each of 47 through 49, use the result of to find the adjoint of ...
 3.2.49: In each of 47 through 49, use the result of to find the adjoint of ...
 3.2.50: For the second order linear equation P(x)y + Q(x)y + R(x)y = 0, sho...
 3.2.51: A second order linear equation P(x)y + Q(x)y + R(x)y = 0 is said to...
Solutions for Chapter 3.2: Solutions of Linear Homogeneous Equations; the Wronskian
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 3.2: Solutions of Linear Homogeneous Equations; the Wronskian
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.2: Solutions of Linear Homogeneous Equations; the Wronskian includes 51 full stepbystep solutions. Since 51 problems in chapter 3.2: Solutions of Linear Homogeneous Equations; the Wronskian have been answered, more than 11660 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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