Advanced book on Mathematics Olympiad

192 3 Real Analysis

Example.Find all continuous functionsf:R→Rsatisfying the equation

f(x)=λ( 1 +x^2 )

[

1 +

∫x

0

f(t)

1 +t^2

dt

]

,

for allx∈R. Hereλis a fixed real number.

Solution.Becausefis continuous, the right-hand side of the functional equation is a
differentiable function; hencefitself is differentiable. Rewrite the equation as

f(x)

1 +x^2

=λ

[

1 +

∫x

0

f(t)

1 +t^2

dt

]

,

and then differentiate with respect toxto obtain

f′(x)( 1 +x^2 )−f(x) 2 x

( 1 +x^2 )^2

=λ

f(x)

1 +x^2

.

We can separate the variables to obtain

f′(x)

f(x)

=λ+

2 x

1 +x^2

,

which, by integration, yields

lnf(x)=λx+ln( 1 +x^2 )+c.

Hencef(x)=a( 1 +x^2 )eλxfor some constanta. Substituting in the original relation,
we obtaina=λ. Therefore, the equation from the statement has the unique solution

f(x)=λ( 1 +x^2 )eλx. 

A first-order differential equation can be written formally as

p(x, y)dx+q(x, y)dy= 0.

Physicists think of the expression on the left as the potential of a two-dimensional force
field, withpandqthexandycomponents of the potential. Mathematicians call this
expression a 1-form. The force field is called conservative if no energy is wasted in
moving an object along any closed path. In this case the differential equation is called
exact. As a consequence of Green’s theorem, the field is conservative precisely when the
exterior derivative
(
∂q
∂x

−

∂p

∂y

)

dxdy