 3.3.1: Find the general solutions of the differential equations in 1 throu...
 3.3.2: Find the general solutions of the differential equations in 1 throu...
 3.3.3: Find the general solutions of the differential equations in 1 throu...
 3.3.4: Find the general solutions of the differential equations in 1 throu...
 3.3.5: Find the general solutions of the differential equations in 1 throu...
 3.3.6: Find the general solutions of the differential equations in 1 throu...
 3.3.7: Find the general solutions of the differential equations in 1 throu...
 3.3.8: Find the general solutions of the differential equations in 1 throu...
 3.3.9: Find the general solutions of the differential equations in 1 throu...
 3.3.10: Find the general solutions of the differential equations in 1 throu...
 3.3.11: Find the general solutions of the differential equations in 1 throu...
 3.3.12: Find the general solutions of the differential equations in 1 throu...
 3.3.13: Find the general solutions of the differential equations in 1 throu...
 3.3.14: Find the general solutions of the differential equations in 1 throu...
 3.3.15: Find the general solutions of the differential equations in 1 throu...
 3.3.16: Find the general solutions of the differential equations in 1 throu...
 3.3.17: Find the general solutions of the differential equations in 1 throu...
 3.3.18: Find the general solutions of the differential equations in 1 throu...
 3.3.19: Find the general solutions of the differential equations in 1 throu...
 3.3.20: Find the general solutions of the differential equations in 1 throu...
 3.3.21: Solve the initial value problems given in 21 through 26.
 3.3.22: Solve the initial value problems given in 21 through 26.
 3.3.23: Solve the initial value problems given in 21 through 26.
 3.3.24: Solve the initial value problems given in 21 through 26.
 3.3.25: Solve the initial value problems given in 21 through 26.
 3.3.26: Solve the initial value problems given in 21 through 26.
 3.3.27: Find general solutions of the equations in 27 through 32. First fin...
 3.3.28: Find general solutions of the equations in 27 through 32. First fin...
 3.3.29: Find general solutions of the equations in 27 through 32. First fin...
 3.3.30: Find general solutions of the equations in 27 through 32. First fin...
 3.3.31: Find general solutions of the equations in 27 through 32. First fin...
 3.3.32: Find general solutions of the equations in 27 through 32. First fin...
 3.3.33: In 33 through 36, one solution of the differential equation is give...
 3.3.34: In 33 through 36, one solution of the differential equation is give...
 3.3.35: In 33 through 36, one solution of the differential equation is give...
 3.3.36: In 33 through 36, one solution of the differential equation is give...
 3.3.37: Find a function y.x/ such that y.4/.x/ D y.3/.x/ for all x and y.0/...
 3.3.38: Solve the initial value problem y.3/ 5y00 C 100y0 500y D 0I y.0/ D ...
 3.3.39: In 39 through 42, find a linear homogeneous constantcoefficient eq...
 3.3.40: In 39 through 42, find a linear homogeneous constantcoefficient eq...
 3.3.41: In 39 through 42, find a linear homogeneous constantcoefficient eq...
 3.3.42: In 39 through 42, find a linear homogeneous constantcoefficient eq...
 3.3.43: (a) Use Eulers formula to show that every complex number can be wri...
 3.3.44: Use the quadratic formula to solve the following equations. Note in...
 3.3.45: Find a general solution of y00 2iy0 C 3y D 0.
 3.3.46: Find a general solution of y00 iy0 C 6y D 0. 47. Find a general sol...
 3.3.47: Find a general solution of y00 D 2 C 2ip3 y.
 3.3.48: Solve the initial value problem y.3/ D yI y.0/ D 1; y0 .0/ D y00.0/...
 3.3.49: Solve the initial value problem y.4/ D y.3/ C y00 C y0 C 2yI y.0/ D...
 3.3.50: The differential equation y00 C .sgn x/y D 0 (25) has the discontin...
 3.3.51: The differential equation y00 C .sgn x/y D 0 (25) has the discontin...
 3.3.52: Make the substitution v D ln x of to find general solutions (for x>...
 3.3.53: Make the substitution v D ln x of to find general solutions (for x>...
 3.3.54: Make the substitution v D ln x of to find general solutions (for x>...
 3.3.55: Make the substitution v D ln x of to find general solutions (for x>...
 3.3.56: Make the substitution v D ln x of to find general solutions (for x>...
 3.3.57: Make the substitution v D ln x of to find general solutions (for x>...
 3.3.58: Make the substitution v D ln x of to find general solutions (for x>...
Solutions for Chapter 3.3: Homogeneous Equations with Constant Coefficients
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 3.3: Homogeneous Equations with Constant Coefficients
Get Full SolutionsChapter 3.3: Homogeneous Equations with Constant Coefficients includes 58 full stepbystep solutions. Since 58 problems in chapter 3.3: Homogeneous Equations with Constant Coefficients have been answered, more than 16523 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.

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

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

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.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

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

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

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.

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.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.