This ODE isn't of Bernoulli type, but it is linear, so we should be able to find an integrating factor to solve it.
[tex]x\dfrac{\mathrm dy}{\mathrm dx}+2y=x^2\log x\implies\dfrac{\mathrm dy}{\mathrm dx}+\dfrac2xy=x\log x[/tex]
The integrating factor will be
[tex]\mu(x)=\exp\left(\displaystyle\int\frac2x\,\mathrm dx\right)=x^2[/tex]
Multiplying both sides of the ODE by the IF gives
[tex]x^2\dfrac{\mathrm dy}{\mathrm dx}+2xy=x^3\log x[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}[x^2y]=x^3\log x[/tex]
[tex]x^2y=\displaystyle\int x^3\log x\,\mathrm dx[/tex]
Integrate the right hand side by parts to get
[tex]x^2y=\dfrac14x^4\log x-\dfrac1{16}x^4+C[/tex]
[tex]y=\dfrac14x^2\log x-\dfrac1{16}x^2+\dfrac C{x^2}[/tex]
