CALCULUS, EARLY TRANS. -WEBASSIGN ACCES
9th Edition
ISBN: 2819260099505
Author: Stewart
Publisher: CENGAGE C
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4. Which substitution would you use to simplify the following integrand? S
a) x = sin
b) x = 2 tan 0
c) x = 2 sec
3√√3
3
x3
5. After making the substitution x = =
tan 0, the definite integral
2
2
3
a) ៖ ស្លឺ sin s
π
- dᎾ
16 0 cos20
b) 2/4 10 cos 20
π
sin30
6
- dᎾ
c)
Π
1 cos³0
3
· de
16 0 sin20
1
x²√x²+4
3
(4x²+9)2
π
d) cos²8
16 0 sin³0
dx
d) x = tan 0
dx simplifies to:
de
6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy?
a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor
7. Evaluate
3x
x+4
- dx
1
a) 3x+41nx + 4 + C b) 31n|x + 4 + C c)
3 ln x + 4+ C d) 3x - 12 In|x + 4| + C
x+4
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?
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