Separate the variables.
[tex]\dfrac{dy}{dx} = x (x - 3) \implies dy = x (x - 3) \, dx[/tex]
Expand the right side.
[tex]dy = (x^2 - 3x) \, dx[/tex]
Integrate both sides. On the right side, use the power rule on each term.
[tex]\displaystyle \int dy = \int (x^2 - 3x) \, dx \implies y = \frac{x^3}3 - \frac{3x^2}2 + C[/tex]
Given that [tex]y=3[/tex] when [tex]x=2[/tex], solve for [tex]C[/tex].
[tex]3 = \dfrac{2^3}3 - \dfrac{3\cdot2^2}2 + C \implies C = \dfrac{19}3[/tex]
Then the particular solution is
[tex]\boxed{y = \dfrac{x^3}3 - \dfrac{3x^2}2 + \dfrac{19}3}[/tex]
