Math Help - [SOLVED] Differential equation

1. [SOLVED] Differential equation

Find the function that satisfies the differential equation
and the condition .

So first of all I found integration factor which is
Then I plugged it in for the general solution formula.

which came out to be

y= 6dt+C

then it came out to be

+C

then I plugged in the values and got C=1 but all the work done wasn't correct

2. When you multiply by the integrating factor, the equation becomes
$e^{-t^{2}}\frac{dy}{dt}-2te^{-t^{2}}y=6t^{2}.$
Integrating, you get
$e^{-t^{2}}y=2t^{3}+C,$ so
$y=e^{t^{2}}(2t^{3}+C),$ etc. .