Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} + 2y \sec^{2} x = 2 \sec^{2} x + 3 \tan x \cdot \sec^{2} x \) such that \( y(0) = \dfrac{5}{4} \). Then \( 12 \left( y\!\left(\dfrac{\pi}{4}\right) - e^{-2} \right) \) is equal to __________________.
