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This ODE isn't of Bernoulli type, but it is linear, so we should be able to find an integrating factor to solve it.
The integrating factor will be
Multiplying both sides of the ODE by the IF gives
![\dfrac{\mathrm d}{\mathrm dx}[x^2y]=x^3\log x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E2y%5D%3Dx%5E3%5Clog%20x)
Integrate the right hand side by parts to get
The integrating factor will be
Multiplying both sides of the ODE by the IF gives
Integrate the right hand side by parts to get