 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 t 2et , 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+ tet y = 0 has y1 and y2 as a f...
 3.2.35: If the differential equation t 2y 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 cons...
 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 e...
 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, show ...
 3.2.51: A second order linear equation P(x)y+ Q(x)y+ R(x)y = 0 is said to b...
Solutions for Chapter 3.2: Solutions of Linear Homogeneous Equations; the Wronskian
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 3.2: Solutions of Linear Homogeneous Equations; the Wronskian
Get Full SolutionsChapter 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 16934 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310.

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.

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.

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

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].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

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

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 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)·(b  Ax) = o.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.