How to solve the following problem: $$\dot{x}=x^3+1.5x^2+0.5x$$
Obviouly, this is a separable nonlinear homogenous first order ODE. So
$$\int\frac{dx}{x^3+1.5x^2+0.5x}=\int dt$$ Then it is a hard work here for integration. Is there any smart way to solve this tedious problem?
Notice that $$x^3+1.5x^2+0.5x= x(x+1)(x+1/2)$$ and by the partial fraction decomposition: $$\frac{1}{x^3+1.5x^2+0.5x}=\frac{2}{x}+\frac{2}{x+1}-\frac{4}{x+1/2}.$$ Therefore after the integration we get $$2\ln\left|\frac{x(x+1)}{(2x+1)^2}\right|=t+c$$ that is $$x(x+1)=Ce^{t/2}(2x+1)^2.$$ Now solve the quadratic equation in $x$. Can you take it from here?
P.S. Don't forget the stationary solutions: $x=0$, $x=-1$ and $x=-1/2$!