Question

Solve differential equation dy/dx=y(y-1)

Answer 1

Solve the separable equation $\frac{dy(x)}{dx} = (y(x) - 1)y(x)$ :

Divide both sides by $(y(x) - 1)y(x)$ :

$$\frac {\frac {d y (x)}{d x}}{(y (x) - 1) y (x)} = 1$$

Integrate both sides with respect to $x$ :

$$\int \frac {\frac {d y (x)}{d x}}{(y (x) - 1) y (x)} d x = \int 1 d x$$

Evaluate the integrals:

$\log (-y(x) + 1) - \log (y(x)) = x + c_1$ , where $c_{1}$ is an arbitrary constant.

Solve for $y(x)$ :

$$y (x) = \frac {1}{e ^ {x + c _ {1}} + 1}$$

Simplify the arbitrary constant:

$$y (x) = \frac {1}{c _ {1} e ^ {x} + 1}$$