Question #41588

Let $P(r) = r^4 - 8r^2 + 16$. Let $r_1, r_2, r_3, r_4$ be the roots of the equation $P(r) = 0$. Let $V_1$ be the vector space over $\mathbb{R}$ spanned by the set of functions $S_1 = \{e^{r_1 x}, e^{r_2 x}, e^{r_3 x}, e^{r_4 x}\}$. Let $V_2$ be the solution space of the differential equation $\frac{d^4y}{dx^4} - 8\frac{d^2y}{dx^2} + 16y = 0$. Let $d_1 = \dim(V_1)$ and $d_2 = \dim(V_2)$. Let $O$ be the order of the lowest order linear homogeneous differential equation with constant coefficients whose solution space is exactly $V_1$. Let $W(x)$ be the Wronskian of the fundamental solutions $e^{2x}, x e^{2x}, e^{-2x}, x e^{-2x}$ of the above differential equation. Find the value of $d_1 \times d_2 \times O + W(0)$.