1

How can I show that the differential equation:

$${dx \over dt }=\sqrt x, x(0)=0$$

has infinitely many solutions.

One solution will be $x={t^2 \over 4}$

Mini_me
  • 2,165

1 Answers1

1

Let $k$ be a positive real number. Let $f: \mathbb{R} \to \mathbb{R}$ be given by: $f(t)=0$ if $t<k$, and $f(t)=\frac{(t-k)^{2}}{4}$ for $t>k$. Now check that:

  • $f'(t)=\sqrt{f(t)}$ for all $t$
  • $f(0)=0$
  • $f$ is continuously differentiable at $t=k$ (at all other points this is obvious)
preferred_anon
  • 17,121