Express $y$ as a function of $x$. The constant $\mathrm{C}$ is a positive number.
\[
4 \ln y=\frac{1}{2} \ln (3 x+1)-\frac{1}{3} \ln (x+2)+\ln C
\]

Step 1 ：Apply the properties of logarithms to simplify the equation: \(4 \ln(y) = \ln\left(\frac{(3x+1)^{1/2}}{(x+2)^{1/3}}\right) + \ln(C)\)

Step 2 ：Simplify to: \(\ln(y^4) = \ln\left(C \frac{(3x+1)^{1/2}}{(x+2)^{1/3}}\right)\)

Step 3 ：Since the logarithms on both sides of the equation are the same, we can equate the arguments of the logarithms: \(y^4 = C \frac{(3x+1)^{1/2}}{(x+2)^{1/3}}\)

Step 4 ：Express y as a function of x by taking the fourth root of both sides: \(y = \left[C \frac{(3x+1)^{1/2}}{(x+2)^{1/3}}\right]^{1/4}\)

Step 5 ：Check if this solution meets the requirements of the problem by substituting y back into the original equation and see if both sides are equal.

Step 6 ：\(\boxed{y = \left[C \frac{(3x+1)^{1/2}}{(x+2)^{1/3}}\right]^{1/4}}\) is the final answer.