1. The problem is to simplify the differential equation $$\frac{dy}{dx} = \frac{1}{3y}$$ and find the implicit solution. 2. Start by separating variables: multiply both sides by $$3y\,dx$$ to get $$3y\,dy = dx$$. 3. Integrate both sides: $$\int 3y\,dy = \int dx$$ 4. The left integral is $$3 \int y\,dy = 3 \cdot \frac{y^2}{2} = \frac{3y^2}{2}$$. The right integral is $$x + C$$, where $$C$$ is the constant of integration. 5. So the implicit solution is: $$\frac{3y^2}{2} = x + C$$ 6. Comparing with the options given, the correct answer is option b. Final answer: $$\frac{3y^2}{2} = x + C$$