Find the general solution of the Euler equation corresponding to the functional $J[q]=\int_{a}^{b} f(t)\sqrt{1+q'^2} dt$ and investigate the special cases $ f(t)=\sqrt{t}$ and $f(t)=t$
What I got to work
$L(t,q,q')=t\sqrt{1+q'^2}$
$L_{q}(t,q,q')-\frac{d}{dt}\left ( L_{q'}(t,q,q') \right)=0$
$L_{q}(t,q,q')=0$
$L_{q'}(t,q,q')=t*\frac{1}{2\sqrt{1+q'^2}}*2q'=\frac{tq'}{\sqrt{1+q'^2}}$
$-\frac{d}{dt}\left ( L_{q'}(t,q,q') \right)=0$
$-\frac{d}{dt}\left (\frac{tq'}{\sqrt{1+q'^2}} \right)=0$
$\frac{q'\sqrt{1+q'^2}-tq'\frac{1}{2\sqrt{q'^2+1}}*2q'}{1+q'^2}=0$
HELP!!
We cannot solve this exercise when we start to get stuck, without answering thank you
From
$$ -\frac{d}{dt}\left (\frac{tq'}{\sqrt{1+q'^2}} \right)=0 $$
we have
$$ \frac{tq'}{\sqrt{1+q'^2}}=C_0 $$
and following
$$ q'=\frac{\pm C_0}{t^2-C_0^2} $$
etc.
NOTE
If $f(t) = \sqrt{t}$ then
$$ q'=\frac{\pm C_0}{t-C_0^2} $$