Extermal curve for specific problems?

Question

I ran into a quiz question last month. how we can find the Extermal curve for following problem.

$$ \int_1^2 \frac {\dot {x}^2}{t^3} dt $$

where $x(1)=2, \ x(2)=17$

Answer 1

If the integrand is $L=t^{-3}\dot{x}^2$ Euler-Lagrange equation is $$ 0 = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}} \right) - \frac{\partial L}{\partial x} = \frac{d}{dt}(t^{-3}\dot{x}). $$ Integrating, $$ A = t^{-3} \dot{x} \\ \dot{x} = At^3 \\ x = A' t^4 + B, $$ and now choose $A'$ and $B$ so the boundary conditions are satisfied, which will be when $A'=B=1$.