332
CHAPTER 38
38.54
Approximate
to three decimal places
For
Hence,
Integrate termwise:
Since this is an alternating series, we look for the least n such that
We get n = 2. Thus, we may use
38.55 Approximate tan l \ to two decimal places.
By Problem 38.36,
for
Therefore,
By the alternating series theorem, we seek the least n for which
We obtain n = 3. Thus, we can use
38.56 Use power series to solve the differential equation y" = 4y with the boundary conditions y = 0, y' = l
when x = 0.
Let
Since y = 0 when x = 0, a 0 = 0. Differentiate:
Since y' — 1
when x = 0, a l = l. Differentiate again:
Since
Therefore,
we get
For odd subscripts,
Hence,
Then,
By Problem 38.44, sinh u =
Hence, y = I sinh 2x.
38.57
Show directly that, if y"=-y, and y' = l and y = 0 when jt = 0, then _y = sin;t.
Let z = dyldx.
Then
Hence,
Since z = l and y=0 when x = 0, K=l. Thus,
Since y = 0 when x = 0, C, = 0. So
Then y=sinx. (If y = -sinjc, then y' = -cosx, and y' = -1 when
38.58 Show that
Let
When x = 0, y = 0. By differentiation,
Hence,
y' = 1 when jc = 0. Further,
-y. Hence, by Problem 38.57, y = sin x.
38.59 Show that
By Problem 38.58,
Differentiate:
In general,