Mathematics for Economists

Calculus of variations

It is a non-linear equation. Assume thatx^0 depends only onxand not on

t.Letu=x^0 ,x^00 =dtdu^0 =du/dx
dx/dt=du/dx
u

1 = xududxu^2

1 +u^2

u

= xdu

dx

1

2

2 u

u^2 + 1

du =^1

x

dx

ln

p

1 +u^2 = ln[xj+C,

p

1 +u^2 =Cx

q x

1 +



x

2

= C