Given the IVP:

$$\frac{dy}{dx} = \sqrt{|y(x)|}$$

$$y(x_0) = 0$$

Show that it has at least two solutions. I really don't know how to do this. I don't know of any necessary conditions for the uniqueness of the solution (if there are any).

Any suggestions? Thanks.

  • $\begingroup$ In this case you don't need any necessary condition for the uniqueness of the solution. You have just to give two different function who satisfies your IVP. $\endgroup$ – Leonhardt von M Apr 7 '15 at 8:20
  • $\begingroup$ One of the solutions is $y = 0$.. Can you find another? $\endgroup$ – mattos Apr 7 '15 at 8:25
  • $\begingroup$ @Mattos, can you? $\endgroup$ – goldenratio Apr 7 '15 at 8:27
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    $\begingroup$ @goldenratio Have you even tried? $\endgroup$ – Git Gud Apr 7 '15 at 8:35
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    $\begingroup$ Another option is to answer it yourself completely, and mark it as best answer. That way, someone else might find the question, and might find it interesting. $\endgroup$ – mickep Apr 7 '15 at 8:39

Define the functions $y_0$ and $y_1$ by $y_0(x) = 0$ for every $x$, $y_1(x) = 0$ for every $x\le x_0$, and $y_1(x)=\frac14(x-x_0)^2$ for every $x>x_0$, then $y_0$ and $y_1$ are two different solutions of this IVP.


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