Mathematics for Economists

Calculus of variations

Example

Find the extremal of the functional

Z 3

1

( 3 tx)xdt, x( 1 )= 1 ,x( 3 )= 4

1

2.

The kernel function is

F



t,x,x



=( 3 tx)x.

Fx^0 = 3 t 2 x,Fx^0 = 0.

The EulerñLagrange equation is 3t 2 x= 0 ,sox= 3 / 2 t.But att= 1

x( 1 )= 3 /2 the Örst boundary condition is not valid. Hence there is no

extremal solution.