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Let $x, y$ be real numbers such that $$(\sqrt{y^{2} - x\,\,}\, - x)(\sqrt{x^{2} + y\,\,}\, - y)=y.$$ Prove $x=-y$. Any suggestion would be appreciated.

Would you please tell me more specifically? I haven't got the picture.

Let $f$ be a solution of the following equation $y''+p(x)y=0$, $p$ is continuous on $\mathbb{R}$ such that $p(x)\leq 0$ for all $x\in\mathbb{R}$. Suppose that $f$ is defined on $[a,+\infty)$, $f(a)>0$, $f'(a)>0$, $a\in\mathbb{R}$ . Prove $f(x)>0$ for all $x\in[a,\infty)$. Any help would be...

Calculate the following integral $$\int\limits_0^1\frac{(1-x)e^x}{x+e^x}dx$$