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 3.9: Give an interval over which f1(x) x2 and f2(x) x x are linearly i...
 3.10: Without the aid of the Wronskian determine whether the given set of...
 3.11: Suppose m1 3, m2 5, and m3 1 are roots of multiplicity one, two, an...
 3.12: Find a CauchyEuler differential equation ax2 ybxycy 0, where a, b, ...
 3.13: In 1328, use the procedures developed in this chapter to find the ...
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 3.29: Write down the form of the general solution y yc yp of the given di...
 3.30: (a) Given that y sin x is a solution of y(4) 2y 11y 2y 10y 0, find ...
 3.31: (a) Write the general solution of the fourthorder DE y(4) 2y y 0 e...
 3.32: Consider the differential equation x2 y (x2 2x)y (x 2)y x3 . Verify...
 3.33: In 3338, solve the given differential equation subject to the indic...
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 3.37: In 3338, solve the given differential equation subject to the indic...
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 3.39: (a) Use a CAS as an aid in finding the roots of the auxiliary equat...
 3.40: Find a member of the family of solutions of xy y !x 0 whose graph i...
 3.41: In 4144, use systematic elimination to solve the given system of d...
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 3.45: A free undamped spring/mass system oscillates with a period of 3 s....
 3.46: A 12pound weight stretches a spring 2 feet. The weight is released...
 3.47: A spring with constant k 2 is suspended in a liquid that offers a d...
 3.48: A 32pound weight stretches a spring 6 inches. The weight moves thr...
 3.49: A series circuit contains an inductance of L 1 h, a capacitance of ...
 3.50: Show that the current i(t) in an LRCseries circuit satisfies the d...
 3.51: Consider the boundaryvalue problem y ly 0, y(0) y(2p), y(0) y(2p)....
 3.52: Sliding Bead A bead is constrained to slide along a frictionless ro...
 3.53: Suppose a mass m lying on a flat, dry, frictionless surface is atta...
 3.54: What is the differential equation of motion in if kinetic friction ...
 3.55: In 55 and 56, use a Greens function to solve the given initialvalu...
 3.56: In 55 and 56, use a Greens function to solve the given initialvalu...
 3.57: Ballistic Pendulum Historically, in order to maintain quality contr...
 3.58: Use the result in to find the muzzle velocity vb when mb 5g, mw 1kg...
 3.59: Use a Maclaurin series to show that a power series solution of the ...
 3.60: Spring Pendulum The rotational form of Newtons second law of motion...
 3.61: The Paris Guns The first mathematically correct theory of projectil...
 3.62: The Paris GunsContinued Mathematically, Galileos model in is perfec...
Solutions for Chapter 3: HigherOrder Differential Equations
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 3: HigherOrder Differential Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 3: HigherOrder Differential Equations includes 62 full stepbystep solutions. Since 62 problems in chapter 3: HigherOrder Differential Equations have been answered, more than 36063 students have viewed full stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

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

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).

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

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

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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 ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Solvable system Ax = b.
The right side b is in the column space of A.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·