 8.2.8.1.27: In 112 find the general solution of the given system.dxdt x 2y
 8.2.8.1.28: In 112 find the general solution of the given system.dxdt 2x 2y
 8.2.8.1.29: In 112 find the general solution of the given system.dxdt 4x 2y
 8.2.8.1.30: In 112 find the general solution of the given system.dxdt 52x 2y
 8.2.8.1.31: In 112 find the general solution of the given system.X X108512Xdy
 8.2.8.1.32: In 112 find the general solution of the given system.X 6321X X10
 8.2.8.1.33: In 112 find the general solution of the given system.dxdt x y z
 8.2.8.1.34: In 112 find the general solution of the given system.dxdt 2x 7y
 8.2.8.1.35: In 112 find the general solution of the given system.X 110123011X
 8.2.8.1.36: In 112 find the general solution of the given system.X 101010101X
 8.2.8.1.37: In 112 find the general solution of the given system.X 13418132140312X
 8.2.8.1.38: In 112 find the general solution of the given system.X 140410226X
 8.2.8.1.39: In 13 and 14 solve the given initialvalue problem.X 121012X, X(0) 35
 8.2.8.1.40: In 13 and 14 solve the given initialvalue problem.X 101121401X, X(...
 8.2.8.1.41: In 15 and 16 use a CAS or linear algebra software as an aid in find...
 8.2.8.1.42: In 15 and 16 use a CAS or linear algebra software as an aid in find...
 8.2.8.1.43: (a) Use computer software to obtain the phase portrait of the syste...
 8.2.8.1.44: Find phase portraits for the systems in 2 and 4. For each system fi...
 8.2.8.1.45: In 1928 find the general solution of the given system.dx dt 3x y
 8.2.8.1.46: In 1928 find the general solution of the given system.dx dt 6x 5y
 8.2.8.1.47: In 1928 find the general solution of the given system.X X 1335X
 8.2.8.1.48: In 1928 find the general solution of the given system.X 12490X X 1
 8.2.8.1.49: In 1928 find the general solution of the given system.dxdt 3x y z
 8.2.8.1.50: In 1928 find the general solution of the given system.dxdt 3x 2y 4z dx
 8.2.8.1.51: In 1928 find the general solution of the given system.X X510402025X
 8.2.8.1.52: In 1928 find the general solution of the given system.X 100031011 X X
 8.2.8.1.53: In 1928 find the general solution of the given system.X X120021010X
 8.2.8.1.54: In 1928 find the general solution of the given system.X 400140014 X X
 8.2.8.1.55: In 29 and 30 solve the given initialvalue problem.X 2146X, X(0) 16
 8.2.8.1.56: In 29 and 30 solve the given initialvalue problem.X 001010100X, X(...
 8.2.8.1.57: Show that the 5 5 matrix has an eigenvalue l1 of multiplicity 5. Sh...
 8.2.8.1.58: Find phase portraits for the systems in 20 and 21. For each system ...
 8.2.8.1.59: In 3344 find the general solution of the given system.dxdt 6x y
 8.2.8.1.60: In 3344 find the general solution of the given system.dxdt x y
 8.2.8.1.61: In 3344 find the general solution of the given system.dxdt 5x y
 8.2.8.1.62: In 3344 find the general solution of the given system.dxdt 4x 5y
 8.2.8.1.63: In 3344 find the general solution of the given system.X X4554X
 8.2.8.1.64: In 3344 find the general solution of the given system.X 1183X X
 8.2.8.1.65: In 3344 find the general solution of the given system.dxdt z
 8.2.8.1.66: In 3344 find the general solution of the given system.dxdt z
 8.2.8.1.67: In 3344 find the general solution of the given system.X X111110201X
 8.2.8.1.68: In 3344 find the general solution of the given system.X 404060104 X X
 8.2.8.1.69: In 3344 find the general solution of the given system.X X250560142X
 8.2.8.1.70: In 3344 find the general solution of the given system.X 211420402 X X
 8.2.8.1.71: In 45 and 46 solve the given initialvalue problem.X 11112211432X, ...
 8.2.8.1.72: In 45 and 46 solve the given initialvalue problem.X 6514X, X(0) 28
 8.2.8.1.73: Find phase portraits for the systems in 36, 37, and 38
 8.2.8.1.74: (a) Solve (2) of Section 7.6 using the first method outlined in the...
 8.2.8.1.75: Solve each of the following linear systems. (a) (b) Find a phase po...
 8.2.8.1.76: Consider the 5 5 matrix given in 31. Solve the system X AX without ...
 8.2.8.1.77: Obtain a Cartesian equation of the curve define parametrically by t...
 8.2.8.1.78: Examine your phase portraits in 47. Under what conditions will the ...
Solutions for Chapter 8.2: Systems of Linear FirstOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 8.2: Systems of Linear FirstOrder Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Since 52 problems in chapter 8.2: Systems of Linear FirstOrder Differential Equations have been answered, more than 20336 students have viewed full stepbystep solutions from this chapter. Chapter 8.2: Systems of Linear FirstOrder Differential Equations includes 52 full stepbystep solutions.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

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.

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

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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