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

Answer

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Answer

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

Steps

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