Final Answer
$\frac{-9}{y}+8\ln\left(y+1\right)-8\ln\left(y\right)=x-6\ln\left(x+6\right)+C_1$
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Step-by-step Solution
Specify the solving method
1
Divide fractions $\frac{1}{\frac{y\left(y+1\right)}{\frac{y+9}{y}}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
$\frac{\frac{y+9}{y}}{y\left(y+1\right)}=\frac{1}{\frac{x+6}{x}}$
2
Divide fractions $\frac{1}{\frac{x+6}{x}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
$\frac{\frac{y+9}{y}}{y\left(y+1\right)}=\frac{x}{x+6}$
3
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
$\int\frac{y+9}{y^2\left(y+1\right)}dy=\int\frac{x}{x+6}dx$
4
Solve the integral $\int\frac{y+9}{y^2\left(y+1\right)}dy$ and replace the result in the differential equation
$\frac{-9}{y}+8\ln\left(y+1\right)-8\ln\left(y\right)=\int\frac{x}{x+6}dx$
5
Solve the integral $\int\frac{x}{x+6}dx$ and replace the result in the differential equation
$\frac{-9}{y}+8\ln\left(y+1\right)-8\ln\left(y\right)=x-6\ln\left(x+6\right)+C_1$
Final Answer
$\frac{-9}{y}+8\ln\left(y+1\right)-8\ln\left(y\right)=x-6\ln\left(x+6\right)+C_1$