ODE with infinitely many solutions

Question

We have that: $ \dot{x}=x^{2/3} $, and $x(0)=0$. This Initial Value Problem has infinitely many solutions (is a problem in my Dynamics Book, so I'm sure about that), but personally I could found only two, $x=t^3/27$ and $x=0$. Normally I would say that a linear combination of the solutions will also be a solution, but in that case we have $x=0$. How can I find another?

Answer 1

Your equation is translation invariant. That means that if $x=f(t)$ is a solution, then $x=f(t-c)$ is also a solution. The way to get infinitely many solutions is by pasting $x=0$ with $x=(t-c)^3/27$ at $x=c>0$. It easy so see that the resulting function is regular and satisfies the equation at all points.

You should get the family of solutions given in MPW's answer in this way (the constant $c_1$ there is related to my $c$ above).

Answer 2

Try Wolfram Alpha. It shows the solution as $$y(x) = \frac{1}{27}\left(3c_1x^2 + 3c_1^2x + c_1^3 + x^3\right)$$ where each value of $c_1$ yields a different solution.