(Solved): A1.2 Consider the initial value problem (IVP): \[ \begin{array}{l} 2 t^{2} y^{\prime \prime}+3 t y ...

A1.2 Consider the initial value problem (IVP): \[ \begin{array}{l} 2 t^{2} y^{\prime \prime}+3 t y^{\prime}-y=0, \quad t \in I \\ y\left(t_{0}\right)=y_{0}, \quad y^{\prime}\left(t_{0}\right)=y_{0}^{\prime} \end{array} \] (a) Find the interval \( I \) such that the IVP in (2) has a unique solution. (b) Show that the function \( y_{1}(t)=t^{-1} \) is a solution of Eq. 2 . (c) Consider the function \( y_{2}=C_{1} y_{1} \) where \( C_{1} \in \mathbb{R} \) is a real non-zero constant. Do the functions \( y_{1}, y_{2} \) form a fundamental set of solutions? Please justify your answer. (d) Using the method of reduction of order, find a solution \( y_{2} \) such as \( y_{1}, y_{2} \) form a fundamental set of solutions.