Advanced High-School Mathematics

304 CHAPTER 5 Series and Differential Equations

(c) The fourth derivative off satisfies the inequality

∣∣

∣f(4)(x)

∣∣

∣≤ 6

for all x in the closed interval [0,2]. Use the Lagrange er-

ror bound on the approximation tof(0) found in part (c) to

explain whyf(0) is negative.

5. (a) Using mathematical induction, together with l’Hˆopital’s rule,
   prove that limx→∞

Pn(x)

ex

= 0 where Pn(x) is a polynomial of

degree n. Conclude that for any polynomial of degree n,

x→±∞lim

Pn(x)

ex^2

= 0.

(b) Show that limx→ 0

P(^1 x)

e

x^12 = 0, whereP is a polynomial. (Lety=^1 x,

and note that asx→ 0 , y→±∞.)

(c) Letf(x) =e−

x^12

,x 6 = 0 and show by induction that

f(n)(x) = Qn

( 1

x

)

e−

x^12

, where Qn is some polynomial (though

not necessarily of degreen).

(d) Conclude from parts (b) and (c) that

limx→ 0

dn

dxn

Å

e−^1 /x

2 ã

= 0

for alln≥0.

(e) What does all of this say about the Maclaurin series fore−^1 /x

2

?

5.5 Differential Equations

In this section we shall primarily considerfirst-order ordinary^23 dif-
ferential equations(ODE), that is, differential equations of the form
y′ =F(x,y). If the functionF is linear in y, then the ODE is called
alinear ordinary differential equation. A linear differential equation
is therefore expressible in the form y′ =p(x)y+q(x), wherep andq
are functions defined on some common domain. In solving an ODE,
we expect that an arbitrary constant will come as the result of an inte-
gration and would be determined by specifying aninitial valueof the

(^23) to be distinguished from “partial” differential equations.