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/dxdx/dt=du/dxu
1 = xududx u^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