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3h
comment Separable ODE and singular solutions
Even in this example, one could have avoided the error if one had integrated with more care: $$ \int_{y_0}^y \frac{dy'}{(1-(y')^2)^{1\over 2}} = \int_{0}^x 2x'\, dx',\qquad \lvert y(x)\rvert<1$$ taking into account that the function that one finds might cease to be a valid solution if touches the singular values $y=\pm 1$. That's because in that case one is dividing by zero in the equation (1). This integration correctly produces the local solutions $y(x)=\sin(x^2+\arcsin y_0)$, valid only for small $x$ (see the main text).
3h
comment Separable ODE and singular solutions
@CABJ: I am glad you liked this answer (you could upvote and/or accept it if you like). Regarding the books, the problem here is that you have a failure of the uniqueness theorem, due to the fact that $(1-y^2)^{1\over 2}$ is not Lipschitz continuous near $y=\pm 1$. This produces the merging of multiple solutions shown above. But when you are Lipschitz continuous in $y$ (and that's almost always the case, in practice), you can separate variables without much worrying.
4h
awarded  Enlightened
5h
awarded  Nice Answer
5h
comment If $T^{*}$ is injective then $T$ is surjective?
Related.
5h
comment Separable ODE and singular solutions
P.S.: This example comes from this discussion on another mathematical forum (in Italian).
5h
revised Separable ODE and singular solutions
added 29 characters in body
5h
answered Separable ODE and singular solutions
11h
revised Open Unit Ball diffeomorphic to the Open Unit Cube
corrected two typos in formulas
14h
revised Separation of variables for the Laplace equation on a disk
improved title + fixed typo
15h
comment Open Unit Ball diffeomorphic to the Open Unit Cube
The transformation $$T_p(x_1, x_2)=(x_1^{2\over p}, x_2^{2\over p})$$ maps the first quadrant of the open unit disk onto the first quadrant of the supercircle of power $p$. When $p$ is big this can be a good approximation of the map you are looking for. Unfortunately you cannot directly pass to the limit as this will only give you a homeomorphism and not a diffeomorphism between the two sets. This is because the unit square has corners while supercircles have not. (This is not an answer but I hope it helps nonetheless)
2d
answered Inequalities and Differentiation
Apr
16
comment Is there a function integrable over the interior of a set but not over the entire set?
@user231951: There must be a notational problem here. Which book are you reading, and what exercise is this? (You'd better put this information in the OP, IMHO)
Apr
16
answered Is there a function integrable over the interior of a set but not over the entire set?
Apr
16
comment Daily exercises to speed up my mental calculations?
Psychology is very important here. The moment you think "I can't do it", you get lost. This is something I see on myself all the time while doing computations. Keep focused and leave all "background noise" away from your brain.
Apr
16
answered Show that this function is weakly differentiable
Apr
16
comment Question about convergence in $L^2$ (revisited)
en.wikipedia.org/wiki/Nemytskii_operator
Apr
16
comment How to integrate $\int{x^a e^{-x} \, dx}$, if a is not an integer?
Are you talking about the indefinite integral, or the integral over $[0, +\infty)$?
Apr
16
comment If $E$ is closed and bounded in a metric space $X$ and $f: E → R$ is continous on $E$ , prove that $f$ is bounded on $E$.
This is false. Not all metric spaces have the Bolzano-Weierstrass property. That's why your book does it on $R$.
Apr
15
comment Locally Lipschitz complex function
Related: math.stackexchange.com/q/328338/8157