# Need help with ODE and Existence and Uniqueness Thm

1. Jun 3, 2006

### 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.

-------------
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?
-------------

I'm pretty sure that the theorem they are referring to is this one,
http://mathworld.wolfram.com/PicardsExistenceTheorem.html
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.

Last edited: Jun 3, 2006
2. Jun 3, 2006

### 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:
$y_{1}(x)=0, y_{2}(x)=\frac{1}{16}((1+x)^{2}-4)^{2}$

Last edited: Jun 3, 2006
3. Jun 3, 2006

### 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 :)

Last edited: Jun 3, 2006