If $y = y(x)$ is the solution of the differential equation $2x^{2} \frac{dy}{dx} - 2xy + 3y^{2} = 0$ such that $y(e) = \frac{e}{3}$,then $y(1)$ is equal to

The solution curve of the differential equation $2 y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx}$,passing through the point $(0, 1)$,is a conic whose vertex lies on the line:

If $y=y(x)$ satisfies the differential equation $8 \sqrt{x}(\sqrt{9+\sqrt{x}}) dy = (\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} dx$ for $x>0$ and $y(0)=\sqrt{7}$,then find $y(256)$.

The particular solution of the differential equation $(y + x \cdot \frac{dy}{dx}) \cdot \sin(xy) = \cos x$ at $x = 0$ is

The equation of the curve passing through the point $\left(0, \frac{\pi}{4}\right)$ and satisfying the differential equation $\left(e^x \tan y\right) dx + \left((1+e^x) \sec^2 y\right) dy = 0$ is given by

The solution of $\cos y \frac{dy}{dx} = e^{x+\sin y} + x^2 e^{\sin y}$ is $f(x) + e^{-\sin y} = C$ ($C$ is an arbitrary real constant), where $f(x)$ is equal to: