The Wronskian & Variation of Parameters · Challenge problem

The functions $t$, $t^2$ and $e^{t}$ are solutions of a third-order equation $y^{\prime\prime\prime}+p_2(t)\,y''+p_1(t)\,y'+p_0(t)\,y=0$ with continuous coefficients on the whole real line. Find $p_2(2)$.