Final Answer
$y=\frac{e^x}{C_1+e^x}$
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Step-by-step Solution
Specify the solving method
1
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
$\frac{1}{y\left(1-y\right)}dy=dx$
2
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
$\int\frac{1}{y\left(1-y\right)}dy=\int1dx$
3
Solve the integral $\int\frac{1}{y\left(1-y\right)}dy$ and replace the result in the differential equation
$\ln\left(y\right)-\ln\left(-y+1\right)=\int1dx$
4
Solve the integral $\int1dx$ and replace the result in the differential equation
$\ln\left(y\right)-\ln\left(-y+1\right)=x+C_0$
5
Find the explicit solution to the differential equation. We need to isolate the variable $y$
$y=\frac{e^x}{C_1+e^x}$
Final Answer
$y=\frac{e^x}{C_1+e^x}$