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 □

Let $y=y(x)$ be the solution of the differential equation $\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0$, $y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal...

The solution curve, of the differential equation $2 y \frac{d y}{d x}+3=5 \frac{d y}{d x}$ , passing through the point (0,1) is a conic, whose...

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $\mathrm{e}^{-\mathrm{a}}+4 \mathrm{a}^2+\mathrm{a}-1$. Then the differential equation,...

If y=y(x) is the solution of the differential equation dydx+2y=sin⁡(2x),y(0)=34, then y8 is equal to :