Asymptotic stability of the zero solution of the equation $x'=x/(1+t)-x^3$

Question

Problem

Prove the zero solution of the following equation is asymptotically stable $$\frac{dx}{dt}=\frac{x}{1+t}-x^3$$

Progress

1. The equation is nonlinear and non-autonomous.
2. Without the nonlinear term $x^3$ we have a separable equation which is easy to solve: $x(t) = A(1+t)$ for any constant $A$. These solutions are not stable; if $A\ne 0$, they move away from the equilibrium solution. Therefore, the nonlinear term is essential for stability; but it makes the equation difficult to solve explicitly. How to proceed?

Answer 1

Hint: Let $V = \dfrac{1}{2}x^2$. Then, $\dfrac{dV}{dt} = x\dfrac{dx}{dt} = \dfrac{x^2}{1+t}-x^4 < 0$ if $|x| > \dfrac{1}{\sqrt{1+t}}$.

What does this tell you about $x(t)$?

Answer 2

This is a Bernoulli ODE. Method of solving : http://mathworld.wolfram.com/BernoulliDifferentialEquation.html