 4.1: In 1 through 16, transform the given differential equation or syste...
 4.2: In 1 through 16, transform the given differential equation or syste...
 4.3: In 1 through 16, transform the given differential equation or syste...
 4.4: In 1 through 16, transform the given differential equation or syste...
 4.5: In 1 through 16, transform the given differential equation or syste...
 4.6: In 1 through 16, transform the given differential equation or syste...
 4.7: In 1 through 16, transform the given differential equation or syste...
 4.8: In 1 through 16, transform the given differential equation or syste...
 4.9: In 1 through 16, transform the given differential equation or syste...
 4.10: In 1 through 16, transform the given differential equation or syste...
 4.11: In 1 through 16, transform the given differential equation or syste...
 4.12: In 1 through 16, transform the given differential equation or syste...
 4.13: In 1 through 16, transform the given differential equation or syste...
 4.14: In 1 through 16, transform the given differential equation or syste...
 4.15: In 1 through 16, transform the given differential equation or syste...
 4.16: In 1 through 16, transform the given differential equation or syste...
 4.17: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.18: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.19: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.20: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.21: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.22: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.23: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.24: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.25: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.26: Use the method of Examples 6, 7, and 8 to nd general solutions of t...
 4.27: (a) Calculate x.t/"2Cy.t/"2 to show that the trajectories of the sy...
 4.28: (a) Beginning with the general solution of the system x0 D!2y, y0 D...
 4.29: First solve Eqs. (20) and (21) for e!t and e2t in terms of x.t/, y....
 4.30: Derive the equationsm1x00 1 D!.k1 Ck2/x1 C k2x2, m2x00 2 D k2x1 ! ....
 4.31: Twoparticleseach ofmass m areattached toastringunder (constant) ten...
 4.32: Three 100gal fermentation vats are connected as indicated in Fig. ...
 4.33: Set up a system of rstorder differential equations for the indicat...
 4.34: Repeat 33, except with the generator replaced with a battery supply...
 4.35: A particle of massm moves in the plane with coordinates .x.t/;y.t//...
 4.36: Suppose that a projectile of mass m moves in a vertical plane in th...
 4.37: Supposethataparticlewithmassmandelectricalchargeq moves in the xyp...
 4.38: Consider the system of two masses and three springs shown in Fig. 4...
 4.39: In 39 through 46, nd the general solution of the system in with the...
 4.40: In 39 through 46, nd the general solution of the system in with the...
 4.41: In 39 through 46, nd the general solution of the system in with the...
 4.42: In 39 through 46, nd the general solution of the system in with the...
 4.43: In 39 through 46, nd the general solution of the system in with the...
 4.44: In 39 through 46, nd the general solution of the system in with the...
 4.45: In 39 through 46, nd the general solution of the system in with the...
 4.46: In 39 through 46, nd the general solution of the system in with the...
 4.47: (a) For the system shown in Fig. 4.2.7, derive the equations of mot...
 4.48: Suppose that the trajectory .x.t/;y.t// of a particle moving in the...
Solutions for Chapter 4: Introduction to Systems of Differential Equations
Full solutions for Differential Equations: Computing and Modeling  5th Edition
ISBN: 9780321816252
Solutions for Chapter 4: Introduction to Systems of Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations: Computing and Modeling, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4: Introduction to Systems of Differential Equations includes 48 full stepbystep solutions. Differential Equations: Computing and Modeling was written by and is associated to the ISBN: 9780321816252. Since 48 problems in chapter 4: Introduction to Systems of Differential Equations have been answered, more than 1909 students have viewed full stepbystep solutions from this chapter.

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

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

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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.

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

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.