[tex]x\dfrac{\mathrm dy}{\mathrm dx}-2y=x^{12}[/tex]
[tex]\dfrac1{x^2}\dfrac{\mathrm dy}{\mathrm dx}-\dfrac2{x^3}y=x^9[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1{x^2}y\right]=x^9[/tex]
[tex]\dfrac1{x^2}y=\displaystyle\int x^9\,\mathrm dx[/tex]
[tex]\dfrac1{x^2}y=\dfrac1{10}x^{10}+C[/tex]
[tex]y=\dfrac1{10}x^{12}+Cx^{-2}[/tex]
With [tex]y(1)=10[/tex], we have
[tex]10=\dfrac1{10}1^{12}+C(1)^{-2}\implies C=\dfrac{99}{10}[/tex]
and so the particular solution is
[tex]y=\dfrac1{10}x^{12}+\dfrac{99}{10x^2}[/tex]
