Mathematics for Economists

Calculus of variations

Example

On what curve can the functional

Z 2

1



x

 2

 2 txdt, x( 1 )= 0 ,x( 2 )= 1

attain an extremum.

The kernel function isF



t,x,x



=



x

 2

 2 tx.The partial derivatives

Fx^0 = 2 t, Fx^0 = 2



x

The EulerñLagrange equation is

 2 t= d

dt

2 x= 2 x, x=t

The solution is

x(t)=t^2 / 2 +c

1 ,x(t)=

t^3

6 +c^1 t+c^2.