2.2.4.1 Solved using first_order_ode_autonomous

Entering first order ode autonomous solver

\[\begin {aligned} x^{\prime }&=k \left (A -n x\right ) \left (M -m x\right ) \end {aligned}\]

Integrating gives

\begin{align*} \int \frac {1}{k \left (-x n +A \right ) \left (-m x +M \right )}d x &= dt\\ -\frac {\ln \left (-m x +M \right )}{k \left (A m -M n \right )}+\frac {\ln \left (-x n +A \right )}{k \left (A m -M n \right )}&= t +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} k \left (-x n +A \right ) \left (-m x +M \right )&= 0 \end{align*}

for \(x\). This is because of dividing by the above earlier. This gives the following singular solution(s), which also has to satisfy the given ODE.

\begin{align*} x&=\frac {A}{n}\\ x&=\frac {M}{m} \end{align*}

Simplifying \(-\frac {\ln \left (M -m x\right )}{k \left (A m -M n \right )}+\frac {\ln \left (A -n x\right )}{k \left (A m -M n \right )} = t +c_1\) gives

\[ \frac {-\ln \left (M -m x\right )+\ln \left (A -n x\right )}{k \left (A m -M n \right )} = t +c_1 \]

Solving for \(x\) from \(\frac {-\ln \left (M -m x\right )+\ln \left (A -n x\right )}{k \left (A m -M n \right )} = t +c_1\) gives

\begin{align*} x &= \frac {A \,{\mathrm e}^{-A c_1 k m -A k m t +M c_1 k n +M k n t}-M}{{\mathrm e}^{-A c_1 k m -A k m t +M c_1 k n +M k n t} n -m} \\ \end{align*}

Summary of solutions found

\begin{align*} x&=\frac {A \,{\mathrm e}^{-A c_1 k m -A k m t +M c_1 k n +M k n t}-M}{{\mathrm e}^{-A c_1 k m -A k m t +M c_1 k n +M k n t} n -m}\\ x&=\frac {A}{n}\\ x&=\frac {M}{m} \end{align*}