= Solution
For a nonzero real solution use the <Bernoulli power substitution> $u=\sqrt[3]y$. Dividing the <differential equation> by $3u^2$ gives $u'+u/3=x/3$. The <integrating factor> $e^{x/3}$ yields
$$
u=x-3+Ce^{-x/3},\qquad
\boxed{y=(x-3+Ce^{-x/3})^3.}
$$
Direct substitution shows this formula also solves the equation at its zeros. Division temporarily discarded the additional solution $\boxed{y\equiv0}$. Since the real power $y^{2/3}$ is not <locally Lipschitz> at zero, the complete real solution family also permits <waiting-time solutions of a sublinear Bernoulli equation>: join a zero interval to a displayed branch at a root of $x-3+Ce^{-x/3}$. At a joining point the branch and its <derivative> are both zero, so the joined function satisfies the original equation. On each open nonzero interval every solution has the displayed form.
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