Solve the differential equation $\frac{dy}{dx}=y\left(1-y\right)$

Rewrite the fraction $\frac{1}{y\left(1-y\right)}$ in $2$ simpler fractions using partial fraction decomposition

Expand the integral $\int\left(\frac{1}{y}+\frac{1}{1-y}\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

Apply the formula: $\int\frac{n}{ax+b}dx$$=\frac{n}{a}\ln\left(ax+b\right)+C$, where $a=-1$, $b=1$, $x=y$ and $n=1$

Divide $1$ by $-1$

The integral of a constant is equal to the constant times the integral's variable

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$

Take the variable outside of the logarithm

Simplifying the logarithm

Simplify $e^{\left(x+C_0\right)}$ applying properties of exponents

Take the reciprocal of both sides of the equation

We can rename $\frac{1}{C_1e^x}$ as other constant

Simplify the fraction $\frac{-y+1}{y}$

Group the terms of the equation

Take the reciprocal of both sides of the equation

Combine $\frac{C_1}{e^x}+1$ in a single fraction

Divide fractions $\frac{1}{\frac{C_1+e^x}{e^x}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$