CALCULUS, EARLY TRANS. -WEBASSIGN ACCES
9th Edition
ISBN: 2819260099505
Author: Stewart
Publisher: CENGAGE C
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Students have asked these similar questions
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.
Knowledge Booster
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