\(y'-y=11te^{2t}\)
\(e^{-t}y'-e^{-t}y=11te^t\)
\(\dfrac{\mathrm d}{\mathrm dt}\left[e^{-t}y\right]=11te^t\)
\(e^{-t}y=11\displaystyle\int te^t\,\mathrm dt\)
\(e^{-t}y=11e^t(t-1)+C\)
Since \(y(0)=1\), you have
\(1=11(-1)+C\implies C=12\)
and so
\(e^{-t}y=11e^t(t-1)+12\)
\(y=11e^{2t}(t-2)+12e^t\)
