Need help with ODE and Existence and Uniqueness Thm

[uart]

Need help with ODE and "Existence and Uniqueness Thm"

I'm currently helping my neice study for her exams and going though last years test there was this one question that I wasn't sure about.

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Considering the initial value problem,

$$\frac{dy}{dx} = f(x,y)$$

Where $$f(x,y) = (1+x) \sqrt{y}$$ and y(1)=0.

Does the "Existence and Uniqueness Theorem" guarantee the existence of a unique solution?
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I'm pretty sure that the theorem they are referring to is this one,
http://mathworld.wolfram.com/PicardsExistenceTheorem.html
See also,
http://www.utpb.edu/scimath/wkfield/mod3/Exuni.htm

I'm thinking that it (the Theorem) doesn't guarantee a unique solution because $$\frac{\partial f}{\partial y}$$ is not well defined at the initial condition y=0.

Unfortunately I'm not too familiar with this "Existence and Uniqueness" theorem. Also the DE does seem to have a well defined solution of y(x)=0 for x>=1, so I'm really not sure. Can anyone help me out here, thanks.

 

[arildno]

you are quite right; the theorem cannot be used here to guarantee a unique solution.
In this case, you have (at least) two solutions fulfilling the initial value problem:
$y_{1}(x)=0, y_{2}(x)=\frac{1}{16}((1+x)^{2}-4)^{2}$

 

[uart]

Thanks for the answer arildno. I can see how you got the two solutions, y=0 by inspection and the other y = 1/16 (x^2 + 2x -3)^2 by seperating and integrating. Yes, now the question makes sense and it nicely shows the theorem in action. Thanks again :)