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
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Intermediate steps
1
Simplify the expression ${0}$
$\frac{\frac{y+9}{y}}{y\left(y+1\right)}=\frac{x}{x+6}$
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2
Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality
$\frac{\frac{y+9}{y}}{y\left(y+1\right)}dy=\frac{x}{x+6}dx$
Intermediate steps
3
Simplify the expression $\frac{\frac{y+9}{y}}{y\left(y+1\right)}dy$
$\frac{y+9}{y^2\left(y+1\right)}dy=\frac{x}{x+6}dx$
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4
Integrate both sides of the differential equation, the left side with respect to
$\int\frac{y+9}{y^2\left(y+1\right)}dy=\int\frac{x}{x+6}dx$
Intermediate steps
5
Solve the integral $\int\frac{y+9}{y^2\left(y+1\right)}dy$ and replace the result in the differential equation
$\frac{-9}{y}+8\ln\left(y+1\right)-8\ln\left(y\right)=\int\frac{x}{x+6}dx$
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Intermediate steps
6
Solve the integral $\int\frac{x}{x+6}dx$ and replace the result in the differential equation
$\frac{-9}{y}+8\ln\left(y+1\right)-8\ln\left(y\right)=x-6\ln\left(x+6\right)+C_1$
Explain this step further
Final Answer
$\frac{-9}{y}+8\ln\left(y+1\right)-8\ln\left(y\right)=x-6\ln\left(x+6\right)+C_1$