Solve dx/dt=kx^n(1-x)^n

$$\frac { d x } { d t } = k x ^ { n } ( 1 - x ) ^ { n }$$

Solve for k

$$\left\{\begin{matrix}k=0\text{, }&\left(x>0\text{ and }x<1\right)\text{ or }\left(n>0\text{ and }x=1\right)\text{ or }\left(Denominator(n)\text{bmod}2=1\text{ and }x>1\right)\text{ or }\left(n>0\text{ and }x=0\right)\text{ or }\left(Denominator(n)\text{bmod}2=1\text{ and }x<0\right)\\k\in \mathrm{R}\text{, }&\left(x=1\text{ or }x=0\right)\text{ and }n>0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$kx^{n}\left(1-x\right)^{n}=\frac{\mathrm{d}}{\mathrm{d}t}(x)$$

The equation is in standard form.

$$\left(x\left(1-x\right)\right)^{n}k=0$$

Divide 0 by x^{n}\left(1-x\right)^{n}.

$$k=0$$

Solve for n

$$\left\{\begin{matrix}n\in \mathrm{R}\text{, }&\left(x>0\text{ or }Denominator(n)\text{bmod}2=1\right)\text{ and }x\neq 1\text{ and }x\neq 0\text{ and }k=0\text{ and }\left(x<1\text{ or }Denominator(n)\text{bmod}2=1\right)\\n>0\text{, }&x=1\text{ or }x=0\end{matrix}\right.$$