Solve x(dy/dx)=y+x^3
when y=3 and x =1
have i done this correct ?
It all looks good until you say $\displaystyle \displaystyle \int P ~dx = -\frac{1}{x} = \ln x$ which makes little sense.
You are trying to find an integrating factor $\displaystyle \displaystyle I = e^{\int -\frac{1}{x}~dx}$ go from there.
As pickslides said, you were good until the middle term of this compound equation: $\displaystyle \displaystyle \int P ~dx = -\frac{1}{x} = \ln x\,.$
It should have been: $\displaystyle \displaystyle \int P ~dx = \int-\frac{1}{x} \,dx= -\ln x\,.$ We ignore the integration constant when finding an integrating factor.
The integrating factor is then: $\displaystyle e^{-ln(x)}\,,$ which can be simplified.