Mathematics for Economists

Calculus of variations

Example

Solve the problem

Z 1

0

tx+



x

 2

dt!min,x( 0 )= 1.

The EulerñLagrange is

0 =Fx^0 = d

dt

Fx^0 = d

dt



t+ 2 x



= 1 + 2 x.

C=t+ 2 
x, x( 0 )= 1

x=C

2 

t

2 , x(^0 )=^1

x(t) 1 =

C

2 t

t^2

4.

x(t)=

t^2

4 +

C

2 t+^1