Chapter 4.7, Problem 35E

Given that $1+t$, $1+2t$, and $1+3t^2$ are solutions to the differential equation $y^{\prime \prime }+p\left(t\right)y^{\prime }+q\left(t\right)y=g\left(t\right)$, find the solution to this equation that satisfies $y\left(1\right)=2$, $y^{\prime }\left(1\right)=0$.