In this Lagrangian problem, action is $$S = \int_{t_1}^{t_2} \sqrt{t}\sqrt{1+\dot{y}^2} \,\,dt$$ where $\dot{y} = dy/dt$ and $t_1$ and $t_2$ are some fixed points.
I tried to solve this problem using Euler-Lagrange:
$$L = \sqrt{t}\sqrt{1+\dot{y}^2},$$ so $\frac{\partial{L}}{\partial y} = 0$ and therefore $\frac{\partial L}{\partial \dot{y}} = K$ where $K$ is some constant.
Now after some substitutions, I get: $$2K\sqrt{t} + C = \sqrt{1+\dot{y}^2}$$ where $C$ is some constant.
We can square LHS and RHS and get: $$(2K\sqrt{t} + C)^2 -1 = \dot{y}^2.$$
And then we can substitute $u = \dot{y}$ and we get:
$$\sqrt{(2K\sqrt{t} + C)^2 -1}\,\,dx = dy.$$
But integrating this results in a weird answer, because I have a solution that says the answer is of form $2\sqrt{A(t-A)}+B$. What should be done to achieve this solution?