Mathematics for Economists

Calculus of variations

This is again an equation with separable variables

c

q

jyjdy = dx

c 1 jyj^3 /^2 = x+c 2

y = c 33

q

(x+c 2 )^2 ,y( 0 )= 1 ,y( 1 )=^3

p

4

1 = c 33

q

c 22 ,^3

p

4 =c 33

q

( 1 +c 2 )^2

There are two solutions

c 3 = c 2 = 1 ,y=^3

q

(x+ 1 )^2

c 3 =^3

p

9 ,c 2 =^1

3

,y=^3

p

93

s

x^1

3

 2

=^3

q

( 3 x 1 )^2.