 4.4.1.97: State some applications that can be modeled by systems of ODEs.
 4.4.1.98: What is population dynamics? Give examples.
 4.4.1.99: How can you transform an ODE into a system of ODEs?
 4.4.1.100: What are qualitative methods for systems? Why are they important?
 4.4.1.101: What is the phase plane? The phase plane method? The phase portrait...
 4.4.1.102: What is a critical point of a system of ODEs? How did we classify t...
 4.4.1.103: What are eigenvalues? What role did they play in this chapter?
 4.4.1.104: What does stability mean in general? In connection with critical po...
 4.4.1.105: What does linearization of a system mean? Give an example.
 4.4.1.106: What is a limit cycle? When may it occur in mechanics?
 4.4.1.107: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.108: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.109: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.110: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.111: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.112: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.113: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.114: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.115: Find a general solution. Determine the kind and stability of the cr...
 4.4.1.116: Find a general solution. (Show the details.) y~ = 3)'2 + 6t y~ = 12...
 4.4.1.117: Find a general solution. (Show the details.))'~ = )'1 + 2.\'2 + e2t
 4.4.1.118: Find a general solution. (Show the details.)
 4.4.1.119: Find a general solution. (Show the details.)
 4.4.1.120: Find a general solution. (Show the details.) y~ = Y1  2Y2  sin ty...
 4.4.1.121: Find a general solution. (Show the details.)
 4.4.1.122: (Mixing problem) Tank Tl in Fig. 99 contains initially 200 gal of w...
 4.4.1.123: (Critical point) What kind of critical point does y' = Ay have if A...
 4.4.1.124: (Network) Find the currents in Fig. 100. where R1 = 0.5 fl, R2 = 0....
 4.4.1.125: (Network) Find the currents in Fig. 10 1 when R = 10 fl, L = 1.25 H...
 4.4.1.126: Detelmine the location and kind of all critical points of the given...
 4.4.1.127: Detelmine the location and kind of all critical points of the given...
 4.4.1.129: Detelmine the location and kind of all critical points of thegiven ...
Solutions for Chapter 4: Systems of ODEs. Phase Plane. Qualitative Methods
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 4: Systems of ODEs. Phase Plane. Qualitative Methods
Get Full SolutionsChapter 4: Systems of ODEs. Phase Plane. Qualitative Methods includes 32 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 32 problems in chapter 4: Systems of ODEs. Phase Plane. Qualitative Methods have been answered, more than 48383 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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 .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.