Consider three vectors $$\vec{a}, \vec{b}, \vec{c}$$. Let $$|\vec{a}|=2,|\vec{b}|=3$$ and $$\vec{a}=\vec{b} \times \vec{c}$$. If $$\alpha \in\left[0, \frac{\pi}{3}\right]$$ is the angle between the vectors $$\vec{b}$$ and $$\vec{c}$$, then the minimum value of $$27|\vec{c}-\vec{a}|^2$$ is equal to:

The differential equation of the family of circles passing through the origin and having centre at the line $$y=x$$ is :

Let $$y=y(x)$$ be the solution of the differential equation

$$\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0.$$

Then the area enclosed by the curve $$f(x)=y(x) \mathrm{e}^{-\frac{1}{\left(1+x^2\right)}}$$ and the line $$y-x=4$$ is ________.

Let the mean and the standard deviation of the probability distribution

be $$\mu$$ and $$\sigma$$, respectively. If $$\sigma-\mu=2$$, then $$\sigma+\mu$$ is equal to ________.