Mathematics for Economists

Calculus of variations

Integrating by parts and using thaty(a)=y(b)= 0

0 =

Zb

a

q(t)y(t)+p(t)y^0 (t)dt=

=

Zb

a

q(t)y(t)dt+

Zb

a

p(t)y^0 (t)dt=

=

Zb

a

q(t)y(t)dt+[p(t)y(t)]ba

Zb

a

p^0 (t)y(t)dt=

=

Zb

a

q(t)y(t)dt

Zb

a

p^0 (t)y(t)dt=

=

Zb

a

q(t)p^0 (t)

y(t)dt.