Express $\mathrm{y}$ as a function of $\mathrm{x}$. The constant $\mathrm{C}$ is a positive \[ 5 \ln y=\frac{1}{2} \ln (4 x+1)-\frac{1}{3} \ln (x+9)+\ln C \]

Step 1 ：\(\ln y=\frac{1}{10} \ln (4 x+1)-\frac{1}{15} \ln (x+9)+\frac{1}{5}\ln C\)

Step 2 ：\(\ln y=\ln \left(\frac{(4x+1)^{1/10}}{(x+9)^{1/15}}\right)+\frac{1}{5}\ln C\)

Step 3 ：\(\ln y=\ln \left(\frac{(4x+1)^{1/10}C^{1/5}}{(x+9)^{1/15}}\right)\)

Step 4 ：\(\boxed{y=\frac{(4x+1)^{1/10}C^{1/5}}{(x+9)^{1/15}}}\)