Mathematics for Economists

Calculus of variations

Example

Find the extremal solution of

Zb

a

y+(y

(^0) )^3
3
dx
The EulerñLagrange equation is
1 =
d
dx

y^0

2

,x+c=

y^0

2

,

y^0 = 

p

x+c.

As there are no boundary conditions

0 = Fy^00

b,y(b),y^0 (b)

=

y^0

2

=b+c

0 = Fy^00

a,y(a),y^0 (a)

=

y^0

2

=a+c

Which has no solution, hence there are no extremal solutions.