Given y1(t)=t2\displaystyle {y}_{{1}}{\left({t}\right)}={t}^{{2}} and y2(t)=t−1\displaystyle {y}_{{2}}{\left({t}\right)}={t}^{{-{{1}}}} satisfy the corresponding homogeneous equation of

t2y′′−2y=−3t3+2,t>0\displaystyle {t}^{{2}}{y}{''}-{2}{y}=-{3}{t}^{{3}}+{2},\quad{t}>{0}

Then the general solution to the non-homogeneous equation can be written as y(t)=c1y1(t)+c2y2(t)+Y(t)\displaystyle {y}{\left({t}\right)}={c}_{{1}}{y}_{{1}}{\left({t}\right)}+{c}_{{2}}{y}_{{2}}{\left({t}\right)}+{Y}{\left({t}\right)}.

Use variation of parameters to find Y(t)\displaystyle {Y}{\left({t}\right)}.

Y(t)\displaystyle {Y}{\left({t}\right)} =  

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