Existence and unicity of solution for a Cauchy problem I have this problem $$\begin{cases} y'=-x^2 y^2+\exp(-y^2)\\ y(0)=0\end{cases}$$
the question is: study the existence and unicity of solution for  this problem on $R=\{(x,y)\in\mathbb{R}^2, |x|\leq \frac12, |y|\leq 1\}$
using two methods.
can someone give me a hint on what is the two methods ?
i have this theorem in the case $y'=f(x,y), y(x_0)=y_0$ if $f$ and $\partial f/\partial y$ are bounded on $R=\{(x,y)\in\mathbb{R}^2, |x-x_0|\leq a, |y-y_0|\leq b\}$ then the Cauchy problem has a unique solution.
in my case $f(x,y)= -x^2 y^2+\exp(-y^2)$
so from the definition of $R$:  $-\frac14\leq-\frac{y^2}{4}\leq -x^2 y^2\leq 0$
and $\exp(-1)\leq\exp(-y^2)\leq 1$
conclusion $-\frac14+\exp(-1)\leq f(x,y)\leq 1 $ so f is bounded
and i fo the same to $\partial f/\partial y=-2x^2 y-2y\exp(-y^2)$
But what is the second methode ?
Edit:
For the second methods I try to calculate  the iterative  sequence and I can't calculate  it
I found
$y_1(x)=x$
$y_2(x)=- \frac15 x^5+\int_0^x \exp(-s^2)ds$
How to do please
Thank you
 A: I would not call it two methods, just two approaches using the same local existence theorem.

*

*As you remarked, the right-side function is smooth on all of $\Bbb R^2$, thus local solutions exist everywhere and can be continued to infinity, be it in $x$ or in $y$. What remains to show is that the solution to the given initial conditions leaves the given box at the sides, not top or bottom.
This follows trivially from $|f(x,y)|\le M=1$, as then $|y(x)-y(0)|\le M|x-0|$ or $|y(x)|\le |x|$.


*Using the details of the proof of the existence theorem directly, the Banach fixed-point theorem can be applied to the Picard iteration if the space $X=C([-a,a]\to[-b,b])$ is mapped inside itself. This is the case if $M$ is a bound on $f$ on $[-a,a]\times[-b,b]$ and the inequality $aM\le b$ holds. This is true for the present values. Additionally a Lipschitz constant in $y$ direction needs to exist on this rectangle, this can be applied in different ways to ensure the existence of a solution over $[-a,a]$.
