The slope of tangent at any point $(x, y)$ on a curve $y=y(x)$ is $\frac{x^2+y^2}{2 x y}, x>0$. If $y(2)=0$, then a value of $x(8)$...

Let $f$ be a differentiable function such that $x^2 f(x)-x=4 \int_0^x t f(t) d t, f(1)=\frac{2}{3}$. Then $18 f(3)$ is equal to

The solution of the differential equation $\left(x^2+y^2\right) d x-5 x y d y=0, y(1)=0$, is :

Let a conic C pass through the point (4,-2) and P(x,y),x≥3, be any point on C. Let the slope of the line touching the conic...

Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$....

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, \mathrm{x}>0$ and $\mathrm{y}\left(\mathrm{e}^{-1}\right)=0$. Then,...

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(1+x^2\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}, \mathrm{y}(1)=0$. Then $\mathrm{y}(0)$ is $\_\_\_\_$ .

Let the solution $y=y(x)$ of the differential equation $\frac{d y}{d x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+$ 10 is equal to □