Page 332 - Schaum's Outline of Differential Equations

CHAP. 32] SECOND-ORDER BOUNDARY-VALUE PROBLEMS 315

2
2
(b) The equation is equivalent to (xy')' + (x + l)y + 'ky = 0; hence p(x) = x, q(x) = x + 1 and w(x) = 1. Since p(x)
is zero at a point in the interval [0, 1], this is not a Sturm-Liouville problem.
(c) Here p(x) = 1/x, q(x) = x, and w(x) = 1. Since p(x) is not continuous in [0, 1], in particular at x = 0, this is not
a Sturm-Liouville problem.
(d) The equation can be rewritten as (y')' + X(l + x)y = 0; hence p(x) = 1, q(x) = 0, and w(x) = 1 +x. This is a
Sturm-Liouville problem.
(e) The equation, in its present form, is not equivalent to Eq. (32.6); this is not a Sturm-Liouville problem.
x
x
x
However, if we first multiply the equation by e~ , we obtain (e y')' + Xe~ y = 0; this is a Sturm-Liouville problem
1
with p(x) = e , q(x) = 0, and w(x) = e~ . x
32.15. Prove that Eq. (32.6) is equivalent to Eq. (32.8) if and only if a' 2(x) = a^x).
Applying the product rule of differentiation to (32.6), we find that

Setting a 2(x) =p(x), cii(x) =p'(x), a Q(x) = q(x), and r(x) = w(x), it follows that (_/), which is (32.6) rewritten, is precisely
(29.8) with a' 2(x) =p'(x) = a^x).
Conversely, if a' 2(x) = cii(x). then (32.8) has the form

which is equivalent to [a 2(;e);y'] + a 0(x)y + hr(x)y = 0. This last equation is precisely (32.6) with p(x) = a 2(x),
q(x) = a a(x), and w(x) = r(x).

32.16. Show that if Eq. (32.8) is multiplied by , the resulting equation is equivalent to
Eq. (32.6).
Multiplying (32.8) by I(x), we obtain

which can be rewritten as

Divide (1) by a 2(x) and then setp(x) = l(x), q(x) = I(x)a a(x)la 2(x) and w(x) = I(x)r(x)la 2(x); the resulting equation is
precisely (32.6). Note that since I(x) is an exponential and since a 2(x) does not vanish, I(x) is positive.

32.17. Transform into Eq. (32.6) by means of the procedure outlined in Problem 32.16.

Here a 2(x) = 1 and cii(x) = 2x; hence a.i(x)la 2(x) = 2x and I(x) = . Multiplying the given differential
equation by I(x), we obtain

which can be rewritten as

This last equation is precisely Eq. (32.6) with p(x) = e x , q(x) = xe x , and w(x)e" .

x
32.18. Transform (x + 2)y" + 4y' + xy + 'ke y = 0 into Eq. (32.6) by means of the procedure outlined in
Problem 32.16.
Here a 2(x) =x + 2 and a^x) = 4; hence a^la^x) = 41 (x + 2) and

Multiplying the given differential equation by I(x), we obtain

which can be rewritten as