Hello!
This is a problem about the general solution of a differential equation.
What we can first do here is separate the variables so that we have the same variable for each side (ex. [tex]dy[/tex] with the [tex]y[/tex] term and [tex]dx[/tex] with the [tex]x[/tex] term).
[tex]\frac{dy}{dx}=\frac{x-1}{3y^2}[/tex]
[tex]3y^2dy=x-1dx[/tex]
Then, we can integrate using the power rule to get rid of the differentiating terms, remember to add the constant of integration, C, to at least one side of the resulting equation.
[tex]y^3=\frac{1}{2}x^2-x+C[/tex]
Then here, we just solve for [tex]y[/tex] and we have our general solution.
[tex]y=\sqrt[3]{\frac{1}{2}x^2-x+C}[/tex]
We can see that answer choice D has an equivalent equation, so answer choice D is the correct answer.
Hope this helps!
