general solution

$$\frac{dy}{dt} = ky \left( M – y \right)$$
$$\frac{dy}{y \left( M – y \right)} = k \ dt$$
$$\int \! \frac{dy}{y \left( M – y \right)} = \int \! k \ dt = kt + C_0$$

In order to integrate the right side of the equation the integrand must be decomposed and integrated by the technique commonly known as “integration by partial fractions.”

$$\frac{1}{y \left( M – y \right)} = \frac{A}{y} + \frac{B}{ \left( M – y \right)}$$
$$1 = A \left( M – y \right) + By = \left( B – A \right)y + AM$$

or

$$0y + 1 = \left( B – A \right)y + AM$$

For this equation to be true the coefficients must be equal. Therefore,

$$B – A = 0$$

and

$$AM=1$$

$$A=B$$
$$A = \frac{1}{M}$$

Therefore, the decomposition is

$$\frac{1}{y \left( M – y \right)} = \frac{1}{My} + \frac{1}{ M\left( M – y \right)}$$

Replacing the integrand with its decomposition and splitting the integral yields

$$\int \! \frac{dy}{y \left( M – y \right)} = \frac{1}{M} \left( \int \! \frac{dy}{y} + \int \! \frac{dy}{M – y} \right)$$

From above it follows

$$\frac{1}{M} \left( \int \! \frac{dy}{y} + \int \! \frac{dy}{M – y} \right) = kt + C_0$$

Now integrating and solving for \( y \)

$$\frac{1}{M} \left( \ln \left| y \right| – \ln \left| M – y \right| + C_1 \right) = kt + C_0$$
$$\ln \left|y \right| – \ln \left| M – y \right| = Mkt + C_2$$
$$\ln \left| \frac{y}{M-y} \right| = Mkt + C_2$$

exponentiate both sides

$$\left| \frac{y}{M-y} \right| = e^{Mkt + C_2} = C_3 e^{Mkt}$$

The absolute value yields two cases.

$$\left| \frac{y}{M-y} \right| = \begin{cases} \ \ \frac{y}{M-y}, & \frac{y}{M-y} \ge 0 \\ -\left( \frac{y}{M-y} \right), & \frac{y}{M-y} < 0 \end{cases}$$

Solving the case for \(y \ge 0\) and assuming that \(y \left( 0 \right) = y_0\)

$$\frac{y_0}{M-y_0} = C_3 e^0 = C_3$$
$$\frac{y}{M-y} = \frac{y_0 e^{Mkt}}{M-y_0}$$
$$y = \frac{y_0 e^{Mkt}}{M-y_0} \left( M – y \right)$$
$$y \left( 1 + \frac{y_0 e^{Mkt}}{M-y_0} \right) = \frac{y_0 M e^{Mkt}}{M-y_0}$$
$$y \left( \frac{M – y_0 + y_0 e^{Mkt}}{M-y_0} \right) = \frac{y_0 M e^{Mkt}}{M-y_0}$$
$$y = \frac{y_0 M e^{Mkt}}{M – y_0 + y_0 e^{Mkt}}$$
$$y = \frac{y_0 M}{ y_0 + \left( M – y_0 \right) e^{-Mkt}}$$