Start with

\[ ye^{3x}\frac{dy}{dx}=x(y+5). \]

Separate the variables:

\[ \frac{y}{y+5}\frac{dy}{dx}=xe^{-3x}. \]

So

\[ \int \frac{y}{y+5}\,dy=\int xe^{-3x}\,dx. \]

For the left-hand side, write

\[ \frac{y}{y+5}=1-\frac{5}{y+5}. \]

Hence

\[ \int \frac{y}{y+5}\,dy=y-5\ln(y+5). \]

For the right-hand side, integrate by parts:

\[ \int xe^{-3x}\,dx=-\frac13xe^{-3x}-\frac19e^{-3x}. \]

Therefore

\[ y-5\ln(y+5)=-\frac13xe^{-3x}-\frac19e^{-3x}+C. \]

Use \(y=0\) when \(x=0\):

\[ 0-5\ln5=-\frac19+C. \]

So

\[ C=\frac19-5\ln5. \]

Therefore

\[ y-5\ln(y+5)=-\frac13xe^{-3x}-\frac19e^{-3x}+\frac19-5\ln5. \]

Equivalently,

\[ y-5\ln\left(\frac{y+5}{5}\right)=\frac19-\left(\frac{x}{3}+\frac19\right)e^{-3x}. \]