If $\frac{dy}{dx} = y + 5$ and $y(0) = 4$, then $y(\log 2)$ is equal to:

The solution of the differential equation $2x \frac{dy}{dx} - y = 0$ with the condition $y(1) = 2$ represents . . . . . . .

The solution of the equation $\frac{dy}{dx} = e^{x - y} + x^2 e^{-y}$ is

The general solution of the differential equation $\frac{dy}{dx} + \frac{1}{\sqrt{1-x^2}} = 0$ is

Find a particular solution satisfying the given condition: $\cos \left(\frac{dy}{dx}\right) = a$ $(a \in R)$; $y = 1$ when $x = 0$.

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 $y(256)$.