Minimization of functional using Euler-Lagrange We've recently started doing Calculus of Variations in my analysis class and we're applying it to minimizing/maximizing functions. So the way we generally were taught to tackle the problem is to first find the Euler-Lagrange equation, solve the differential equation, then check concavity/convexity to ensure uniqueness. I'm having some trouble on the following question: (note: y with the circle thing on top means y')

Problem 4. Solve the minimization problem
  $$
\min \int_1^2 \left(y^2 + 2t\dot y y + 4 t^2 {\dot y}^2\right) dt , \; y(1) = 3, \; y(2)=2
$$

My attempt:

I can't find where I'm going wrong because I'm ending up with a differential equation whose solutions (when I solve the characteristic equations) don't involve t at all, which is problematic. Any help at all would be great! :)

Here 
$$
L(y,\dot y,t) = y^2+2t y\dot y +4t^2 \dot y^2
$$
$$
\frac{\partial L}{\partial y}-\frac{d}{dt}\frac{\partial L}{\partial \dot y} = 8t^2\ddot y+16t \dot y = 0
$$
or
$$
t\ddot y + 2\dot y = 0
$$
now making $z = \dot y \Rightarrow t\dot z + 2 z = 0 \Rightarrow z = C_0 t^{-2}\Rightarrow y = -C_0 t^{-1}+C_1$
etc.
