If $$y^2=a x^2+b x+c$$, where $$a, b, c$$ are constants, then $$y^3 \frac{d^2 y}{d x^2}$$ is equal to

The equation of a circle that passes through the origin and cut off intercepts $$-2$$ and 3 on the $$\mathrm{X}$$-axis and $$\mathrm{Y}$$-axis respectively is

Let

$$f(x)\matrix{ { = |x| + 3,} & {if\,x \le - 3} \cr { = - 2x,} & {if\, - 3 < x < 3} \cr { = 6x - 2,} & {if\,x \ge 3} \cr } $$, then

The particular solution of the diffrential equation $$y(1+\log x)=\left(\log x^x\right) \frac{d y}{d x}$$, when $$y(e)=e^2$$ is