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
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...
TNX in advance...
An initial value problem like this ...
$\displaystyle y^{'} = 2\cdot \sqrt {|y|}$, $\displaystyle y(0)=0$ (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 $\displaystyle y^{'}$ is a function of the $\displaystyle y$ only and that means that if $\displaystyle \varphi(x)$ is solution of (1), then $\displaystyle \varphi(x+\alpha)$ being $\displaystyle \alpha$ an arbitrary constant is also solution of (1). In particular we can observe that solution of (1) is...
$\displaystyle \varphi(x) = x^{2}, x>0$
$\displaystyle \varphi(x)=0, x\le 0$ (2)
... so that any $\displaystyle \varphi (x + \alpha), \alpha<0 $ is also solution of (1)...
Kind regards
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