 Solution

For a nonzero real solution use the Bernoulli power substitution $u=\sqrt[3]{y}$. Dividing the differential equation by $3u^2$ gives $u^{\prime }+u/3=x/3$. The integrating factor $e^{x/3}$ yieldsDirect substitution shows this formula also solves the equation at its zeros. Division temporarily discarded the additional solution $\boxed{y\equiv 0}$. 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 $x-3+C{e}^{-x/3}$. 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.