Solution to nonlinear ODE with square root

Question

How do I solve the following equation? $\dot{x}=\sqrt{x^{2}-\frac{2}{3}x^{3}}$ with $x(0)=0$? I'm guessing I have to work with $dt=\frac{dx}{\sqrt{x^{2}-\frac{2}{3}x^{3}}}$ and integrate in [0,t'] $\Leftrightarrow$ [0,x'] but I'm having problems around $x=0$

Answer 1

Think of your previous question: you know that the answer will be of the form $x(t) = \frac{a}{\text{cosh}^2 t/2}$. Then, what is $x(0)$? It also helps to sketch the graph of $\frac{1}{\text{cosh}^2 t/2}$. What do you notice? Do you see why $x(0) = 0$ yields the trivial solution, as @Did pointed out?

So, given that you have to take a nonzero initial condition $x(0) = x_0 \neq 0$, separation of variables yields \begin{equation} t = \int_{x_0}^x \frac{1}{\sqrt{\xi^2 - \frac{2}{3} \xi^3}}\text{d}\xi. \end{equation} To calculate the integral, it's useful to rewrite the integrand as $\frac{1}{\xi \sqrt{1-\frac{2}{3}\xi}}$ and introduce the new variable $\eta = \sqrt{1-\frac{2}{3}\xi}$, such that $\xi = \frac{3}{2}(1-\eta^2)$, and thus $\text{d} \xi = - 3 \eta \,\text{d} \eta$. To calculate the new integral, look here for inspiration.