If $\frac{dy}{dx} = y + 3$ and $y(0) = 2$,then find the value of $y(\log 2)$.

If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9\sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \, dy = (1+2 \sin y) \, dx$ for $x > 0$, and $y(256)=\frac{\pi}{2}$, $y(49)=\alpha$, then $2 \sin \alpha$ is equal to:

The solution of the differential equation $x \cos y \, dy = (x e^x \log x + e^x) \, dx$ is

Let $y=y(x)$ be the solution of the differential equation $\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y$,with $y(0)=0$. If $y\left(-\frac{2}{3} \log _{e} 2\right)=\alpha \log _{e} 2$,then the value of $\alpha$ is equal to:

The slope of a curve at any point is inversely proportional to twice the $y$-coordinate of that point. If the curve passes through $(4, 3)$,then the equation of the curve is:

Given that the slope of the tangent to a curve $y=y(x)$ at any point $(x, y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$,then its equation is