Given $y''+4ty'+(2+4t^{2})y=t^{2}e^{-t^{2}}$, with $y_1(t)=e^{-t^{2}}$ and $y_2(t)=te^{-t^{2}}$ as solutions to the homogeneous equation, determine:
The complementary solution, $y_c(t)$, the particular solution, $y_p(t)$, using variation of parameters, and form the general solution, $y(t)$.
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