Final answer to the problem
$y=\frac{e^x}{C_1+e^x}$
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Step-by-step Solution
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- Scegliere un'opzione
- Equazione differenziale esatta
- Equazione differenziale lineare
- Equazione differenziale separabile
- Equazione differenziale omogenea
- Prodotto di binomi con termine comune
- Metodo FOIL
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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
Apply the formula: $b\cdot dy=dx$$\to \int bdy=\int1dx$, where $b=\frac{1}{y\left(1-y\right)}$
$\int\frac{1}{y\left(1-y\right)}dy=\int1dx$
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 to the problem
$y=\frac{e^x}{C_1+e^x}$
