Mathematics for Economists

Calculus of variations

0 =ψ^0 ( 0 )=φ^0 a(x(a))y(a)+φ^0 b(x(b))y(b)+

Zb

a

Fx^0 y+Fx^0 y dt.

Again integrating by parts

Zb

a

Fx^0 y dt =

h

Fx^0 y

ib

a

Zb

a

d

dt

Fx^0 ydt=

=Fx^0 y(b)Fx^0 y(a)

Zb

a

d

dtF

^0

xydt.

Hence

0 =

Zb

a



Fx^0 

d

dtF

^0

x



ydt+

+



φ^0 b(x(b))+Fx^0 



b,x(b),x(b)



y(b)+

+



φ^0 a(x(a))Fx^0



a,x(a),x(a)



y(a).