[tex]\dfrac{\mathrm dy}{\mathrm dt}+\dfrac t2y=8t[/tex]
[tex]e^{t^2/2}\dfrac{\mathrm dy}{\mathrm dt}+\dfrac t2e^{t^2/2}y=8te^{t^2/2}[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dt}\left[e^{t^2/2}y\right]=8te^{t^2/2}[/tex]
[tex]e^{t^2/2}y=8\displaystyle\int te^{t^2/2}\,\mathrm dt[/tex]
[tex]e^{t^2/2}y=8e^{t^2/2}+C[/tex]
[tex]y=8+Ce^{-t^2/2}[/tex]
With [tex]y(0)=9[/tex], we get
[tex]9=8+Ce^0\implies C=1[/tex]
and so the particular solution must be
[tex]y=8+e^{-t^2/2}[/tex]
