Finding extremal of a fixed end point problem. Optimisation

Question

I want to find the extremal of the fix-end point problem

$\int_1^2 \frac{\dot{x}^2}{t^3}$ with $x(1)=2,x(2)=17$

First I check the euler-lagrange equation is equal to $0$.

We have: $0-\frac{d}{dt}(\frac{2\dot{x}}{t^3})= \frac{6\dot{x}}{t^4}$

How am I getting this to equate to zero here?

Answer 1

You need to differentiate $\dot x$ with respect to $t$ as well. Of course, it's easier to say $\dot x=ct^3$ for some constant $c$.