Advanced book on Mathematics Olympiad

3.4 Equations with Functions as Unknowns 199

Iterating, we eventually obtain

yiv(x)=−

∫∞

0

e−t

(^2) / 2
cos
x^2
2 t^2
dt=−y(x),
which proves that indeedysatisfies the differential equationyiv+y=0. The general
solution to this differential equation is
y(x)=e
√x
2

(

C 1 cos

x

√

2

+C 2 sin

x

√

2

)

+e−

√x

2

(

C 3 cos

x

√

2

+C 4 sin

x

√

2

)

.

To find which particular solution is the integral in question, we look at boundary values.
To compute these boundary values we refer to Section 3.3.2, the one on multivariable
integral calculus. We recognize thaty( 0 )=

∫∞

0 e

−t^2 / (^2) dtis a Gaussian integral equal
to
√π
2 ,y

′( 0 )=−∫∞

0 sin

u^2

2 duis a Fresnel integral equal to−

√π

2 ,y

′′( 0 )=0, while

y′′′( 0 )=

∫∞

0 cos

u^2

2 duis yet another Fresnel integral equal to

√π
2. We find thatC^1 =
C 2 =C 4 =0 andC 3 =

√π

2. The value of the integral from the statement is therefore

y(x)=

√

π

2

e−

√x

(^2) cos√x
2

. 

An alternative approach is to view the integral as the real part of a (complex) Gaussian
integral.
We leave the following examples to the reader.

567.Show that both functions

y 1 (x)=

∫∞

0

e−tx

1 +t^2

dt and y 2 (x)=

∫∞

0

sint

t+x

dt

satisfy the differential equationy′′+y=^1 x. Prove that these two functions are

equal.

568.Letfbe a real-valued continuous nonnegative function on[ 0 , 1 ]such that

f(t)^2 ≤ 1 + 2

∫t

0

f(s)ds, for allt∈[ 0 , 1 ].

Show thatf(t)≤ 1 +tfor everyt∈[ 0 , 1 ].

569.Letf:[ 0 , 1 ]→Rbe a continuous function withf( 0 )=f( 1 )=0. Assume that
f′′exists on( 0 , 1 )andf′′(x)+ 2 f′(x)+f(x)≥0 for allx∈( 0 , 1 ). Prove that
f(x)≤0 for allx∈[ 0 , 1 ].

570.Does there exist a continuously differentiable functionf : R→ Rsatisfying
f(x) >0 andf′(x)=f(f(x))for everyx∈R?