Calculus
Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 7.8, Problem 56PS

a)

To determine

To prove: that sech2x+tanh2x=1 .

b)

To determine

To prove: that coth2xcsch2x=1

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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.
2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?
Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.

Chapter 7 Solutions

Calculus

Ch. 7.1 - Prob. 11PSCh. 7.1 - Prob. 12PSCh. 7.1 - Prob. 13PSCh. 7.1 - Prob. 14PSCh. 7.1 - Prob. 15PSCh. 7.1 - Prob. 16PSCh. 7.1 - Prob. 17PSCh. 7.1 - Prob. 18PSCh. 7.1 - Prob. 19PSCh. 7.1 - Prob. 20PSCh. 7.1 - Prob. 21PSCh. 7.1 - Prob. 22PSCh. 7.1 - Prob. 23PSCh. 7.1 - Prob. 24PSCh. 7.1 - Prob. 25PSCh. 7.1 - Prob. 26PSCh. 7.1 - Prob. 27PSCh. 7.1 - Prob. 28PSCh. 7.1 - Prob. 29PSCh. 7.1 - Prob. 30PSCh. 7.1 - Prob. 31PSCh. 7.1 - Prob. 32PSCh. 7.1 - Prob. 33PSCh. 7.1 - Prob. 34PSCh. 7.1 - Prob. 35PSCh. 7.1 - Prob. 36PSCh. 7.1 - Prob. 37PSCh. 7.1 - Prob. 38PSCh. 7.1 - Prob. 39PSCh. 7.1 - Prob. 40PSCh. 7.1 - Prob. 41PSCh. 7.1 - Prob. 42PSCh. 7.1 - Prob. 43PSCh. 7.1 - Prob. 44PSCh. 7.1 - Prob. 45PSCh. 7.1 - Prob. 46PSCh. 7.1 - Prob. 47PSCh. 7.1 - Prob. 48PSCh. 7.1 - Prob. 49PSCh. 7.1 - Prob. 50PSCh. 7.1 - Prob. 51PSCh. 7.1 - Prob. 52PSCh. 7.1 - Prob. 53PSCh. 7.1 - Prob. 54PSCh. 7.1 - Prob. 55PSCh. 7.1 - Prob. 56PSCh. 7.1 - Prob. 57PSCh. 7.1 - Prob. 58PSCh. 7.1 - Prob. 59PSCh. 7.1 - Prob. 60PSCh. 7.2 - Prob. 1PSCh. 7.2 - Prob. 2PSCh. 7.2 - Prob. 3PSCh. 7.2 - Prob. 4PSCh. 7.2 - Prob. 5PSCh. 7.2 - Prob. 6PSCh. 7.2 - Prob. 7PSCh. 7.2 - Prob. 8PSCh. 7.2 - Prob. 9PSCh. 7.2 - Prob. 10PSCh. 7.2 - Prob. 11PSCh. 7.2 - Prob. 12PSCh. 7.2 - Prob. 13PSCh. 7.2 - Prob. 14PSCh. 7.2 - Prob. 15PSCh. 7.2 - Prob. 16PSCh. 7.2 - Prob. 17PSCh. 7.2 - Prob. 18PSCh. 7.2 - Prob. 19PSCh. 7.2 - Prob. 20PSCh. 7.2 - Prob. 21PSCh. 7.2 - Prob. 22PSCh. 7.2 - Prob. 23PSCh. 7.2 - Prob. 24PSCh. 7.2 - Prob. 25PSCh. 7.2 - Prob. 26PSCh. 7.2 - Prob. 27PSCh. 7.2 - Prob. 28PSCh. 7.2 - Prob. 29PSCh. 7.2 - Prob. 30PSCh. 7.2 - Prob. 31PSCh. 7.2 - Prob. 32PSCh. 7.2 - Prob. 33PSCh. 7.2 - Prob. 34PSCh. 7.2 - Prob. 35PSCh. 7.2 - Prob. 36PSCh. 7.2 - Prob. 37PSCh. 7.2 - Prob. 38PSCh. 7.2 - Prob. 39PSCh. 7.2 - Prob. 40PSCh. 7.2 - Prob. 41PSCh. 7.2 - Prob. 42PSCh. 7.2 - Prob. 43PSCh. 7.2 - Prob. 44PSCh. 7.2 - Prob. 45PSCh. 7.2 - Prob. 46PSCh. 7.2 - Prob. 47PSCh. 7.2 - Prob. 48PSCh. 7.2 - Prob. 49PSCh. 7.2 - Prob. 50PSCh. 7.2 - Prob. 51PSCh. 7.2 - Prob. 52PSCh. 7.2 - Prob. 53PSCh. 7.2 - Prob. 54PSCh. 7.2 - Prob. 55PSCh. 7.2 - Prob. 56PSCh. 7.2 - Prob. 57PSCh. 7.2 - Prob. 58PSCh. 7.2 - Prob. 59PSCh. 7.2 - Prob. 60PSCh. 7.3 - Prob. 1PSCh. 7.3 - Prob. 2PSCh. 7.3 - Prob. 3PSCh. 7.3 - Prob. 4PSCh. 7.3 - Prob. 5PSCh. 7.3 - Prob. 6PSCh. 7.3 - Prob. 7PSCh. 7.3 - Prob. 8PSCh. 7.3 - Prob. 9PSCh. 7.3 - Prob. 10PSCh. 7.3 - Prob. 11PSCh. 7.3 - Prob. 12PSCh. 7.3 - Prob. 13PSCh. 7.3 - Prob. 14PSCh. 7.3 - Prob. 15PSCh. 7.3 - Prob. 16PSCh. 7.3 - Prob. 17PSCh. 7.3 - Prob. 18PSCh. 7.3 - Prob. 19PSCh. 7.3 - Prob. 20PSCh. 7.3 - Prob. 21PSCh. 7.3 - Prob. 22PSCh. 7.3 - Prob. 23PSCh. 7.3 - Prob. 24PSCh. 7.3 - Prob. 25PSCh. 7.3 - Prob. 26PSCh. 7.3 - Prob. 27PSCh. 7.3 - Prob. 28PSCh. 7.3 - Prob. 29PSCh. 7.3 - Prob. 30PSCh. 7.3 - Prob. 31PSCh. 7.3 - Prob. 32PSCh. 7.3 - Prob. 33PSCh. 7.3 - Prob. 34PSCh. 7.3 - Prob. 35PSCh. 7.3 - Prob. 36PSCh. 7.3 - Prob. 37PSCh. 7.3 - Prob. 38PSCh. 7.3 - Prob. 39PSCh. 7.3 - Prob. 40PSCh. 7.3 - Prob. 41PSCh. 7.3 - Prob. 42PSCh. 7.3 - Prob. 43PSCh. 7.3 - Prob. 44PSCh. 7.3 - Prob. 45PSCh. 7.3 - Prob. 46PSCh. 7.3 - Prob. 47PSCh. 7.3 - Prob. 48PSCh. 7.3 - Prob. 49PSCh. 7.3 - Prob. 50PSCh. 7.3 - Prob. 51PSCh. 7.3 - Prob. 52PSCh. 7.3 - Prob. 53PSCh. 7.3 - Prob. 54PSCh. 7.3 - Prob. 55PSCh. 7.3 - Prob. 56PSCh. 7.3 - Prob. 57PSCh. 7.3 - Prob. 58PSCh. 7.3 - Prob. 59PSCh. 7.3 - Prob. 60PSCh. 7.4 - Prob. 1PSCh. 7.4 - Prob. 2PSCh. 7.4 - Prob. 3PSCh. 7.4 - Prob. 4PSCh. 7.4 - Prob. 5PSCh. 7.4 - Prob. 6PSCh. 7.4 - Prob. 7PSCh. 7.4 - Prob. 8PSCh. 7.4 - Prob. 9PSCh. 7.4 - Prob. 10PSCh. 7.4 - Prob. 11PSCh. 7.4 - Prob. 12PSCh. 7.4 - Prob. 13PSCh. 7.4 - Prob. 14PSCh. 7.4 - Prob. 15PSCh. 7.4 - Prob. 16PSCh. 7.4 - Prob. 17PSCh. 7.4 - Prob. 18PSCh. 7.4 - Prob. 19PSCh. 7.4 - Prob. 20PSCh. 7.4 - Prob. 21PSCh. 7.4 - Prob. 22PSCh. 7.4 - Prob. 23PSCh. 7.4 - Prob. 24PSCh. 7.4 - Prob. 25PSCh. 7.4 - Prob. 26PSCh. 7.4 - Prob. 27PSCh. 7.4 - Prob. 28PSCh. 7.4 - Prob. 29PSCh. 7.4 - Prob. 30PSCh. 7.4 - Prob. 31PSCh. 7.4 - Prob. 32PSCh. 7.4 - Prob. 33PSCh. 7.4 - 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Prob. 25PSCh. 7.5 - Prob. 26PSCh. 7.5 - Prob. 27PSCh. 7.5 - Prob. 28PSCh. 7.5 - Prob. 29PSCh. 7.5 - Prob. 30PSCh. 7.5 - Prob. 31PSCh. 7.5 - Prob. 32PSCh. 7.5 - Prob. 33PSCh. 7.5 - Prob. 34PSCh. 7.5 - Prob. 35PSCh. 7.5 - Prob. 36PSCh. 7.5 - Prob. 37PSCh. 7.5 - Prob. 38PSCh. 7.5 - Prob. 39PSCh. 7.5 - Prob. 40PSCh. 7.5 - Prob. 41PSCh. 7.5 - Prob. 42PSCh. 7.5 - Prob. 43PSCh. 7.5 - Prob. 44PSCh. 7.5 - Prob. 45PSCh. 7.5 - Prob. 46PSCh. 7.5 - Prob. 47PSCh. 7.5 - Prob. 48PSCh. 7.5 - Prob. 49PSCh. 7.5 - Prob. 50PSCh. 7.5 - Prob. 51PSCh. 7.5 - Prob. 52PSCh. 7.5 - Prob. 53PSCh. 7.5 - Prob. 54PSCh. 7.5 - Prob. 55PSCh. 7.5 - Prob. 56PSCh. 7.5 - Prob. 57PSCh. 7.5 - Prob. 58PSCh. 7.5 - Prob. 59PSCh. 7.5 - Prob. 60PSCh. 7.6 - Prob. 1PSCh. 7.6 - Prob. 2PSCh. 7.6 - Prob. 3PSCh. 7.6 - Prob. 4PSCh. 7.6 - Prob. 5PSCh. 7.6 - Prob. 6PSCh. 7.6 - Prob. 7PSCh. 7.6 - Prob. 8PSCh. 7.6 - Prob. 9PSCh. 7.6 - Prob. 10PSCh. 7.6 - Prob. 11PSCh. 7.6 - Prob. 12PSCh. 7.6 - Prob. 13PSCh. 7.6 - Prob. 14PSCh. 7.6 - Prob. 15PSCh. 7.6 - 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Prob. 7PSCh. 7.7 - Prob. 8PSCh. 7.7 - Prob. 9PSCh. 7.7 - Prob. 10PSCh. 7.7 - Prob. 11PSCh. 7.7 - Prob. 12PSCh. 7.7 - Prob. 13PSCh. 7.7 - Prob. 14PSCh. 7.7 - Prob. 15PSCh. 7.7 - Prob. 16PSCh. 7.7 - Prob. 17PSCh. 7.7 - Prob. 18PSCh. 7.7 - Prob. 19PSCh. 7.7 - Prob. 20PSCh. 7.7 - Prob. 21PSCh. 7.7 - Prob. 22PSCh. 7.7 - Prob. 23PSCh. 7.7 - Prob. 24PSCh. 7.7 - Prob. 25PSCh. 7.7 - Prob. 26PSCh. 7.7 - Prob. 27PSCh. 7.7 - Prob. 28PSCh. 7.7 - Prob. 29PSCh. 7.7 - Prob. 30PSCh. 7.7 - Prob. 31PSCh. 7.7 - Prob. 32PSCh. 7.7 - Prob. 33PSCh. 7.7 - Prob. 34PSCh. 7.7 - Prob. 35PSCh. 7.7 - Prob. 36PSCh. 7.7 - Prob. 37PSCh. 7.7 - Prob. 38PSCh. 7.7 - Prob. 39PSCh. 7.7 - Prob. 40PSCh. 7.7 - Prob. 41PSCh. 7.7 - Prob. 42PSCh. 7.7 - Prob. 43PSCh. 7.7 - Prob. 44PSCh. 7.7 - Prob. 45PSCh. 7.7 - Prob. 46PSCh. 7.7 - Prob. 47PSCh. 7.7 - Prob. 48PSCh. 7.7 - Prob. 49PSCh. 7.7 - Prob. 50PSCh. 7.7 - Prob. 51PSCh. 7.7 - Prob. 52PSCh. 7.7 - Prob. 53PSCh. 7.7 - Prob. 54PSCh. 7.7 - Prob. 55PSCh. 7.7 - Prob. 56PSCh. 7.7 - Prob. 57PSCh. 7.7 - Prob. 58PSCh. 7.7 - Prob. 59PSCh. 7.7 - Prob. 60PSCh. 7.8 - Prob. 1PSCh. 7.8 - Prob. 2PSCh. 7.8 - Prob. 3PSCh. 7.8 - Prob. 4PSCh. 7.8 - Prob. 5PSCh. 7.8 - Prob. 6PSCh. 7.8 - Prob. 7PSCh. 7.8 - Prob. 8PSCh. 7.8 - Prob. 9PSCh. 7.8 - Prob. 10PSCh. 7.8 - Prob. 11PSCh. 7.8 - Prob. 12PSCh. 7.8 - Prob. 13PSCh. 7.8 - Prob. 14PSCh. 7.8 - Prob. 15PSCh. 7.8 - Prob. 16PSCh. 7.8 - Prob. 17PSCh. 7.8 - Prob. 18PSCh. 7.8 - Prob. 19PSCh. 7.8 - Prob. 20PSCh. 7.8 - Prob. 21PSCh. 7.8 - Prob. 22PSCh. 7.8 - Prob. 23PSCh. 7.8 - Prob. 24PSCh. 7.8 - Prob. 25PSCh. 7.8 - Prob. 26PSCh. 7.8 - Prob. 27PSCh. 7.8 - Prob. 28PSCh. 7.8 - Prob. 29PSCh. 7.8 - Prob. 30PSCh. 7.8 - Prob. 31PSCh. 7.8 - Prob. 32PSCh. 7.8 - Prob. 33PSCh. 7.8 - Prob. 34PSCh. 7.8 - Prob. 35PSCh. 7.8 - Prob. 36PSCh. 7.8 - Prob. 37PSCh. 7.8 - Prob. 38PSCh. 7.8 - Prob. 39PSCh. 7.8 - Prob. 40PSCh. 7.8 - Prob. 41PSCh. 7.8 - Prob. 42PSCh. 7.8 - Prob. 43PSCh. 7.8 - Prob. 44PSCh. 7.8 - Prob. 45PSCh. 7.8 - Prob. 46PSCh. 7.8 - Prob. 47PSCh. 7.8 - Prob. 48PSCh. 7.8 - 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