# Thread: Simple question in ODE

1. ## Simple question in ODE

Given this ODE :
y'=2*sqrt(|y|) , y(0)=0 ...
Can we find two different soloutions around (0,0) ? If there are, find them... If there are no two different soloutions around (0,0) - explain why...

Help is needed! TNX

2. Dear WannaBe,

I need some further details to try this problem. What do you mean by,
y(0)=0 ?

And what is "around (0,0)"?

3. Hey there Sudharaka,
The problem is an initial value ODE problem. i.e an equation with an initial value condition... In this one, the equation is: y'=2*sqrt(|y|) and the soloution must be a function y(x) that for x=0: y(x)=0 -> y(0)=0...It's just an initial value condition for the equation...

When I say Around (0,0) I mean that:
"Can we find two different soloutions in some neighborhood around (0,0) ?"

Hope you'll be able to help me now... btw-I'll be delighted if you'll be able to help me in the other question I've posted about ODE's...

4. Dear WannaBe,
Please see the attachment below. From that it is evident that this equation have two solutions for all x values. If you have any problems or if you can't read the attachment please feel free to contact me.

Hope this helps.

5. An initial value problem like this ...

$y^{'} = 2\cdot \sqrt {|y|}$, $y(0)=0$ (1)

... in general admits a single solution if the so called 'Lipschitz conditions' are satisfied...

Initial value problem - Wikipedia, the free encyclopedia

... what is not the case of (1). If we observe the (1) we note that $y^{'}$ is a function of the $y$ only and that means that if $\varphi(x)$ is solution of (1), then $\varphi(x+\alpha)$ being $\alpha$ an arbitrary constant is also solution of (1). In particular we can observe that solution of (1) is...

$\varphi(x) = x^{2}, x>0$

$\varphi(x)=0, x\le 0$ (2)

... so that any $\varphi (x + \alpha), \alpha<0$ is also solution of (1)...

Kind regards

$\chi$ $\sigma$

6. TNX a lot!