(Solved): 2. [6] a) [2] Show that \( y_{2}(x)=y_{1}(x) \int \frac{e^{-\int P(x) d x}}{\left[y_{1}(x)\right]^ ...

2. [6] a) [2] Show that \( y_{2}(x)=y_{1}(x) \int \frac{e^{-\int P(x) d x}}{\left[y_{1}(x)\right]^{2}} d x \) is linearly independent from \( y_{1}(x) \). b) [4] Consider the variable coefficient LSOODE \( x^{2} y^{\prime \prime}-\left(2 x^{3}+3 x\right) y^{\prime}+\left(2 x^{2}+3\right) y=0 \). Show that this equation has a solution of the form \( y=x^{m} \), by determining \( m \), and then construct the general solution.