Simplify differential equation

[salvomic]

hi all,
I would like resolve this differential equation (Bernoulli) with Prime:
y'=t^3y^2+2ty, t, y directly (without substitution).
\( y'=t^{3}y^{2}+2ty \)
My book give general solution: y(t)=1/(ce^(t^2)-(2+t^2))
\( y(t)=\frac{1}{ce^{t^{2}}-(2+t^{2}) } \)
Prime give (e^(-t^2)*(-½*e^(t^2)*(t^2-1)+G_0)^-1.
\( e^{-t^{2}}(-\frac{1}{2}e^{t^{2}}(t^{2}-1+G_0)^{-1} \)
I cannot simplify it or collect and so on to get the same result as the book.
Any hint to get the work with calculator?

Thank you.