Advanced book on Mathematics Olympiad

198 3 Real Analysis

564.Letnbe a positive integer. Show that the equation

( 1 −x^2 )y′′−xy′+n^2 y= 0

admits as a particular solution annth-degree polynomial.

565.Find the one-to-one, twice-differentiable solutionsyto the equation

d^2 y

dx^2

+

d^2 x

dy^2

= 0.

566.Show that all solutions to the differential equationy′′+exy=0 remain bounded
asx→∞.

3.4.4 Problems Solved with Techniques of Differential Equations.....

In this section we illustrate how tricks of differential equations can offer inspiration when
one is tackling problems from outside this field.

Example.Letf:[ 0 ,∞)→Rbe a twice-differentiable function satisfyingf( 0 )≥ 0
andf′(x) > f (x)for allx>0. Prove thatf(x) >0 for allx>0.

Solution.To solve this problem we use an integrating factor. The inequality

f′(x)−f(x) > 0

can be “integrated’’ after being multiplied bye−x. It simply says that the derivative of
the functione−xf(x)is strictly positive on( 0 ,∞). This function is therefore strictly
increasing on[ 0 ,∞). So forx>0 we havee−xf(x)>e−^0 f( 0 )=f( 0 )≥0, which
then impliesf(x) >0, as desired. 

Example.Compute the integral

y(x)=

∫∞

0

e−t

(^2) / 2
cos
x^2
2 t^2
dt.
Solution.We will show that the functiony(x)satisfies the ordinary differential equation
yiv+y=0. To this end, we compute
y′(x)=

∫∞

0

e−t

(^2) / 2
sin
x^2
2 t^2

·

−x

t^2

dt=−

∫∞

0

e−x

(^2) / 2 u 2
sin
u^2
2
du
and
y′′(x)=−

∫∞

0

e−x

(^2) / 2 u 2
sin
u^2
2

·

−x

u^2

du=

∫∞

0

e−t

(^2) / 2
sin
x^2
2 t^2
dt.