# Solving $dx/dt=0.2x^2left(1-x/3right)$ for the initial condition $x(0)=x_0$

I am trying to solve the system $$frac{dx}{dt}=0.2x^2left(1-frac{x}{3}right),$$ if $$x(0)=x_0$$. This is a separable ODE, and my attempt is below.
begin{align} frac{dx}{dt}&=0.2x^2left(1-frac{x}{3}right) \ intfrac{dx}{x^2(3-x)}&=frac{1}{15}int dt \ int frac{1/9}{x}+frac{1/3}{x^2}+frac{1/3}{3-x} dx&=frac{1}{15}int dt \ frac{1}{9}ln(x)-frac{1}{3x}-frac{1}{3}ln(3-x)&=frac{t}{15} +C, Cinmathbb{R}. end{align}
However, I am unsure of how to proceed.

Mathematics Asked on November 21, 2021

This is one of the implicit exuation which means it can be written in a form $$F(x,y)=0$$. As you can see, you cannot do anything more meaningful. All this informations tell us that your solution which you have written at the end is final. Remember that we not always can have solution in a form $$x(t)=ldots.$$

Answered by Novice on November 21, 2021

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