Differential Equations with Boundary-Value Problems (MindTap Course List)
Differential Equations with Boundary-Value Problems (MindTap Course List)
9th Edition
ISBN: 9781305965799
Author: Dennis G. Zill
Publisher: Cengage Learning
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Differential Equations
Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…
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Chapter 4.5, Problem 26E

In Problems 15–26 find a linear differential operator that annihilates the given function.

26. e − x sin x − e 2 x cos x

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Chapter 4 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7ECh. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...
Ch. 4.1 - (a) Use the family in Problem 1 to find a solution...Ch. 4.1 - Prob. 12ECh. 4.1 - In Problems 13 and 14 the given two-parameter...Ch. 4.1 - In Problems 13 and 14 the given two-parameter...Ch. 4.1 - In Problems 1522 determine whether the given set...Ch. 4.1 - In Problems 1522 determine whether the given set...Ch. 4.1 - In Problems 1522 determine whether the given set...Ch. 4.1 - In Problems 1522 determine whether the given set...Ch. 4.1 - In Problems 1522 determine whether the given set...Ch. 4.1 - Prob. 20ECh. 4.1 - In Problems 1522 determine whether the given set...Ch. 4.1 - In Problems 1522 determine whether the given set...Ch. 4.1 - In Problems 2330 verify that the given functions...Ch. 4.1 - Prob. 24ECh. 4.1 - In Problems 2330 verify that the given functions...Ch. 4.1 - In Problems 2330 verify that the given functions...Ch. 4.1 - In Problems 2330 verify that the given functions...Ch. 4.1 - Prob. 28ECh. 4.1 - Prob. 29ECh. 4.1 - In Problems 2330 verify that the given functions...Ch. 4.1 - In Problems 31–34 verify that the given...Ch. 4.1 - Prob. 32ECh. 4.1 - In Problems 3134 verify that the given...Ch. 4.1 - In Problems 31–34 verify that the given...Ch. 4.1 - (a) Verify that and are, respectively, particular...Ch. 4.1 - Prob. 36ECh. 4.1 - Prob. 37ECh. 4.1 - Suppose that y1 = ex and y2 = ex are two solutions...Ch. 4.1 - (a) Verify that y1 = x3 and y2 = |x|3 are linearly...Ch. 4.1 - Is the set of functions f1(x) = ex+2, f2(x) = ex3...Ch. 4.1 - Prob. 41ECh. 4.1 - Prob. 42ECh. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - Prob. 6ECh. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 116 the indicated function y1(x) is a...Ch. 4.2 - In Problems 1720 the indicated function y1(x) is a...Ch. 4.2 - In Problems 1720 the indicated function y1(x) is a...Ch. 4.2 - In Problems 1720 the indicated function y1(x) is a...Ch. 4.2 - In Problems 1720 the indicated function y1(x) is a...Ch. 4.2 - Prob. 21ECh. 4.2 - In Problems 21 and 22 the indicated function y1(x)...Ch. 4.2 - Prob. 23ECh. 4.2 - Prob. 24ECh. 4.3 - In Problems 114 find the general solution of the...Ch. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - In Problems 114 find the general solution of the...Ch. 4.3 - Prob. 8ECh. 4.3 - In Problems 114 find the general solution of the...Ch. 4.3 - Prob. 10ECh. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Prob. 13ECh. 4.3 - In Problems 114 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - Prob. 20ECh. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - Prob. 22ECh. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - In Problems 1528 find the general solution of the...Ch. 4.3 - Prob. 28ECh. 4.3 - In Problems 2936 solve the given initial-value...Ch. 4.3 - In Problems 2936 solve the given initial-value...Ch. 4.3 - In Problems 2936 solve the given initial-value...Ch. 4.3 - Prob. 32ECh. 4.3 - Prob. 33ECh. 4.3 - Prob. 34ECh. 4.3 - Prob. 35ECh. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.3 - Prob. 38ECh. 4.3 - Prob. 39ECh. 4.3 - In Problems 3740 solve the given boundary-value...Ch. 4.3 - In Problems 41 and 42 solve the given problem...Ch. 4.3 - In Problems 41 and 42 solve the given problem...Ch. 4.3 - Prob. 43ECh. 4.3 - Prob. 44ECh. 4.3 - Prob. 45ECh. 4.3 - Prob. 46ECh. 4.3 - Prob. 47ECh. 4.3 - Prob. 48ECh. 4.3 - In Problems 4958 find a homogeneous linear...Ch. 4.3 - Prob. 50ECh. 4.3 - Prob. 51ECh. 4.3 - Prob. 52ECh. 4.3 - Prob. 53ECh. 4.3 - Prob. 54ECh. 4.3 - Prob. 55ECh. 4.3 - Prob. 56ECh. 4.3 - Prob. 57ECh. 4.3 - Prob. 58ECh. 4.3 - Two roots of a cubic auxiliary equation with real...Ch. 4.3 - Find the general solution of 2 + 7y + 4y 4y 0 if...Ch. 4.3 - Find the general solution of y + 6y + y 34y = 0...Ch. 4.3 - To solve y(4) + y = 0, we must find the roots of...Ch. 4.3 - Prob. 63ECh. 4.3 - Prob. 64ECh. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - Prob. 9ECh. 4.4 - In Problems 1–26 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - Prob. 12ECh. 4.4 - In Problems 1–26 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - Prob. 20ECh. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - In Problems 126 solve the given differential...Ch. 4.4 - Prob. 27ECh. 4.4 - Prob. 28ECh. 4.4 - Prob. 29ECh. 4.4 - In Problems 2736 solve the given initial-value...Ch. 4.4 - In Problems 2736 solve the given initial-value...Ch. 4.4 - Prob. 32ECh. 4.4 - In Problems 2736 solve the given initial-value...Ch. 4.4 - In Problems 2736 solve the given initial-value...Ch. 4.4 - In Problems 2736 solve the given initial-value...Ch. 4.4 - Prob. 36ECh. 4.4 - Prob. 37ECh. 4.4 - In Problems 3740 solve the given boundary-value...Ch. 4.4 - Prob. 39ECh. 4.4 - In Problems 3740 solve the given boundary-value...Ch. 4.4 - In Problems 41 and 42 solve the given...Ch. 4.4 - In Problems 41 and 42 solve the given...Ch. 4.4 - Consider the differential equation ay + by + cy =...Ch. 4.4 - Prob. 44ECh. 4.4 - In Problems 4548 without solving, match a solution...Ch. 4.4 - In Problems 4548 without solving, match a solution...Ch. 4.4 - In Problems 4548 without solving, match a solution...Ch. 4.4 - Prob. 48ECh. 4.5 - In Problems 110 write the given differential...Ch. 4.5 - In Problems 110 write the given differential...Ch. 4.5 - In Problems 110 write the given differential...Ch. 4.5 - Prob. 4ECh. 4.5 - In Problems 110 write the given differential...Ch. 4.5 - Prob. 6ECh. 4.5 - Prob. 7ECh. 4.5 - Prob. 8ECh. 4.5 - In Problems 110 write the given differential...Ch. 4.5 - In Problems 110 write the given differential...Ch. 4.5 - In Problems 1114 verify that the given...Ch. 4.5 - In Problems 1114 verify that the given...Ch. 4.5 - In Problems 11-14 verify that the given...Ch. 4.5 - In Problems 11-14 verify that the given...Ch. 4.5 - In Problems 15-26 find a linear differential...Ch. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.5 - Prob. 18ECh. 4.5 - Prob. 19ECh. 4.5 - Prob. 20ECh. 4.5 - In Problems 15-26 find a linear differential...Ch. 4.5 - In Problems 1526 find a linear differential...Ch. 4.5 - Prob. 23ECh. 4.5 - In Problems 1526 find a linear differential...Ch. 4.5 - In Problems 1526 find a linear differential...Ch. 4.5 - In Problems 1526 find a linear differential...Ch. 4.5 - In Problems 27-34 find linearly independent...Ch. 4.5 - In Problems 27-34 find linearly independent...Ch. 4.5 - In Problems 27-34 find linearly independent...Ch. 4.5 - Prob. 30ECh. 4.5 - Prob. 31ECh. 4.5 - Prob. 32ECh. 4.5 - In Problems 2734 find linearly independent...Ch. 4.5 - Prob. 34ECh. 4.5 - In Problems 35-64 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - Prob. 39ECh. 4.5 - Prob. 40ECh. 4.5 - Prob. 41ECh. 4.5 - Prob. 42ECh. 4.5 - Prob. 43ECh. 4.5 - Prob. 44ECh. 4.5 - Prob. 45ECh. 4.5 - Prob. 46ECh. 4.5 - Prob. 47ECh. 4.5 - In Problems 35-64 solve the given differential...Ch. 4.5 - Prob. 49ECh. 4.5 - Prob. 50ECh. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - Prob. 52ECh. 4.5 - Prob. 53ECh. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - In Problems 3564 solve the given differential...Ch. 4.5 - Prob. 60ECh. 4.5 - Prob. 61ECh. 4.5 - Prob. 62ECh. 4.5 - Prob. 63ECh. 4.5 - Prob. 64ECh. 4.5 - Prob. 65ECh. 4.5 - Prob. 66ECh. 4.5 - Prob. 67ECh. 4.5 - Prob. 68ECh. 4.5 - Prob. 69ECh. 4.5 - In Problems 6572 solve the given initial-value...Ch. 4.5 - Prob. 71ECh. 4.5 - Prob. 72ECh. 4.5 - Prob. 73ECh. 4.6 - In Problems 118 solve each differential equation...Ch. 4.6 - In Problems 118 solve each differential equation...Ch. 4.6 - In Problems 118 solve each differential equation...Ch. 4.6 - Prob. 4ECh. 4.6 - In Problems 1-18 solve each differential equation...Ch. 4.6 - Prob. 6ECh. 4.6 - In Problems 1-18 solve each differential equation...Ch. 4.6 - In Problems 118 solve each differential equation...Ch. 4.6 - In Problems 118 solve each differential equation...Ch. 4.6 - In Problems 118 solve each differential equation...Ch. 4.6 - Prob. 11ECh. 4.6 - Prob. 12ECh. 4.6 - In Problems 1-18 solve each differential equation...Ch. 4.6 - In Problems 1-18 solve each differential equation...Ch. 4.6 - In Problems 1-18 solve each differential equation...Ch. 4.6 - Prob. 16ECh. 4.6 - In Problems 1-18 solve each differential equation...Ch. 4.6 - Prob. 18ECh. 4.6 - In Problems 19-22 solve each differential equation...Ch. 4.6 - In Problems 19-22 solve each differential equation...Ch. 4.6 - In Problems 19-22 solve each differential equation...Ch. 4.6 - In Problems 19-22 solve each differential equation...Ch. 4.6 - Prob. 23ECh. 4.6 - In Problems 23-26 proceed as in Example 3 and...Ch. 4.6 - In Problems 23-26 proceed as in Example 3 and...Ch. 4.6 - In Problems 23-26 proceed as in Example 3 and...Ch. 4.6 - In Problems 27 and 28 the indicated functions are...Ch. 4.6 - In Problems 27 and 28 the indicated functions are...Ch. 4.6 - Prob. 29ECh. 4.6 - In Problems 2932 solve the given third-order...Ch. 4.6 - Prob. 31ECh. 4.6 - Prob. 32ECh. 4.6 - In Problems 33 and 34 discuss how the methods of...Ch. 4.6 - Prob. 34ECh. 4.6 - Prob. 35ECh. 4.6 - Find the general solution of x4y+x3y4x2y=1 given...Ch. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - Prob. 3ECh. 4.7 - Prob. 4ECh. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - Prob. 6ECh. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - Prob. 8ECh. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - Prob. 10ECh. 4.7 - Prob. 11ECh. 4.7 - Prob. 12ECh. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - Prob. 15ECh. 4.7 - Prob. 16ECh. 4.7 - In Problems 118 solve the given differential...Ch. 4.7 - Prob. 18ECh. 4.7 - In Problems 19-24 solve the given partial equation...Ch. 4.7 - Prob. 20ECh. 4.7 - Prob. 21ECh. 4.7 - Prob. 22ECh. 4.7 - In Problems 19-24 solve the given partial equation...Ch. 4.7 - In Problems 1924 solve the given differential...Ch. 4.7 - In Problems 2530 solve the given initial-value...Ch. 4.7 - Prob. 26ECh. 4.7 - Prob. 27ECh. 4.7 - In Problems 25-30 solve the given initial-value...Ch. 4.7 - Prob. 29ECh. 4.7 - In Problems 25-30 solve the given initial-value...Ch. 4.7 - In Problems 3136 use the substitution x = et to...Ch. 4.7 - Prob. 32ECh. 4.7 - In Problems 3136 use the substitution x = et to...Ch. 4.7 - In Problems 3136 use the substitution x = et to...Ch. 4.7 - Prob. 35ECh. 4.7 - In Problems 3136 use the substitution x = et to...Ch. 4.7 - Prob. 37ECh. 4.7 - In Problems 37 and 38 use the substitution t = x...Ch. 4.7 - Prob. 39ECh. 4.7 - Prob. 40ECh. 4.7 - Prob. 41ECh. 4.7 - Prob. 42ECh. 4.7 - Prob. 43ECh. 4.7 - Prob. 44ECh. 4.7 - Prob. 45ECh. 4.7 - Prob. 46ECh. 4.7 - Bending of a Circular Plate In the analysis of the...Ch. 4.7 - Prob. 48ECh. 4.8 - In Problems 16 proceed as in Example 1 to find a...Ch. 4.8 - In Problems 16 proceed as in Example 1 to find a...Ch. 4.8 - Prob. 3ECh. 4.8 - Prob. 4ECh. 4.8 - Prob. 5ECh. 4.8 - Prob. 6ECh. 4.8 - Prob. 7ECh. 4.8 - In Problems 712 proceed as in Example 2 to find...Ch. 4.8 - Prob. 9ECh. 4.8 - Prob. 10ECh. 4.8 - Prob. 11ECh. 4.8 - Prob. 12ECh. 4.8 - Prob. 13ECh. 4.8 - Prob. 14ECh. 4.8 - Prob. 15ECh. 4.8 - Prob. 16ECh. 4.8 - Prob. 17ECh. 4.8 - In Problems 13-18 proceed as in Example 3 to find...Ch. 4.8 - Prob. 19ECh. 4.8 - Prob. 20ECh. 4.8 - Prob. 21ECh. 4.8 - Prob. 22ECh. 4.8 - In Problems 19-30 proceed as in Example 5 to find...Ch. 4.8 - Prob. 24ECh. 4.8 - Prob. 25ECh. 4.8 - Prob. 26ECh. 4.8 - Prob. 27ECh. 4.8 - Prob. 28ECh. 4.8 - Prob. 29ECh. 4.8 - Prob. 30ECh. 4.8 - Prob. 31ECh. 4.8 - In Problems 31-34 proceed as in Example 6 to find...Ch. 4.8 - Prob. 33ECh. 4.8 - Prob. 34ECh. 4.8 - Prob. 35ECh. 4.8 - Prob. 36ECh. 4.8 - Prob. 37ECh. 4.8 - Prob. 38ECh. 4.8 - Prob. 39ECh. 4.8 - Prob. 40ECh. 4.8 - Prob. 41ECh. 4.8 - In Problems 39-44 proceed as in Examples 7 and 8...Ch. 4.8 - Prob. 43ECh. 4.8 - Prob. 44ECh. 4.9 - In Problems 120 solve the given system of...Ch. 4.9 - Prob. 2ECh. 4.9 - Prob. 3ECh. 4.9 - Prob. 4ECh. 4.9 - Prob. 5ECh. 4.9 - Prob. 6ECh. 4.9 - Prob. 7ECh. 4.9 - In Problems 120 solve the given system of...Ch. 4.9 - Prob. 9ECh. 4.9 - Prob. 10ECh. 4.9 - Prob. 11ECh. 4.9 - Prob. 12ECh. 4.9 - Prob. 13ECh. 4.9 - Prob. 14ECh. 4.9 - Prob. 15ECh. 4.9 - Prob. 16ECh. 4.9 - Prob. 17ECh. 4.9 - In Problems 120 solve the given system of...Ch. 4.9 - Prob. 19ECh. 4.9 - Prob. 20ECh. 4.9 - Prob. 21ECh. 4.9 - Prob. 22ECh. 4.9 - Projectile Motion A projectile shot from a gun has...Ch. 4.9 - Prob. 24ECh. 4.9 - Prob. 25ECh. 4.10 - Prob. 1ECh. 4.10 - Prob. 2ECh. 4.10 - In Problems 3-6 the dependent variable y is...Ch. 4.10 - Prob. 4ECh. 4.10 - In Problems 3-6 the dependent variable y is...Ch. 4.10 - Prob. 6ECh. 4.10 - In Problems 710 the independent variable x is...Ch. 4.10 - Prob. 8ECh. 4.10 - Prob. 9ECh. 4.10 - Prob. 10ECh. 4.10 - Prob. 11ECh. 4.10 - In Problems 11 and 12 solve the given...Ch. 4.10 - Prob. 13ECh. 4.10 - Prob. 14ECh. 4.10 - Prob. 15ECh. 4.10 - In Problems 15 and 16 show that the substitution u...Ch. 4.10 - In Problems 17–20 proceed as in Example 3 and...Ch. 4.10 - Prob. 18ECh. 4.10 - Prob. 19ECh. 4.10 - Prob. 20ECh. 4.10 - In calculus the curvature of a curve that is...Ch. 4.10 - Prob. 22ECh. 4.10 - Discuss how the method of reduction of order...Ch. 4.10 - Prob. 24ECh. 4.10 - Prob. 25ECh. 4 - Answer Problems 110 without referring back to the...Ch. 4 - Answer Problems 110 without referring back to the...Ch. 4 - Answer Problems 110 without referring back to the...Ch. 4 - Prob. 4RECh. 4 - Prob. 5RECh. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Prob. 8RECh. 4 - Prob. 9RECh. 4 - If y1 = ex and y2 = ex are solutions of...Ch. 4 - Prob. 11RECh. 4 - Without the aid of the Wronskian, determine...Ch. 4 - Suppose m1 = 3, m2 = 5, and m3 = 1 are roots of...Ch. 4 - Prob. 14RECh. 4 - In Problems 15 and 16 find a homogeneous...Ch. 4 - Prob. 16RECh. 4 - In Problems 1732 use the procedures developed in...Ch. 4 - Prob. 18RECh. 4 - Prob. 19RECh. 4 - Prob. 20RECh. 4 - In Problems 1732 use the procedures developed in...Ch. 4 - Prob. 22RECh. 4 - Prob. 23RECh. 4 - Prob. 24RECh. 4 - Prob. 25RECh. 4 - Prob. 26RECh. 4 - Prob. 27RECh. 4 - Prob. 28RECh. 4 - Prob. 29RECh. 4 - Prob. 30RECh. 4 - Prob. 31RECh. 4 - Prob. 32RECh. 4 - In Problems 33 and 34 write down the form of the...Ch. 4 - Prob. 34RECh. 4 - Prob. 35RECh. 4 - Prob. 36RECh. 4 - Prob. 37RECh. 4 - Prob. 38RECh. 4 - Prob. 39RECh. 4 - Prob. 40RECh. 4 - Prob. 41RECh. 4 - Prob. 42RECh. 4 - Prob. 43RECh. 4 - Prob. 44RECh. 4 - Prob. 46RECh. 4 - Prob. 47RECh. 4 - In Problems 47-50 use systematic elimination to...Ch. 4 - Prob. 49RECh. 4 - Prob. 50RE
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