If $y(x)$ is the solution of the differential equation $x dy - (y^2 - 4y) dx = 0$ for $x > 0$ and $y(1) = 2$,and the slope of the curve $y = y(x)$ is never zero,then the value of $10y(\sqrt{2})$ is . . . .

If a curve passes through the point $(1,1)$ and at any point $(x, y)$ on the curve,the product of the slope of its tangent and $x$ coordinate of the point is equal to the $y$ coordinate of the point,then the curve also passes through the point

The solution of the differential equation $\frac{dy}{dx} = \cos(x + y)$ is...

If $\frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y} \log_e 2}$ and $y(0) = 0$,then for $y = 1$,the value of $x$ lies in the interval:

If $y = f(x)$ is a monotonically increasing function such that $(\frac{dy}{dx})^2 = 6 - \frac{dy}{dx}$ and $y(0) = 5$, then $y(3) = ...$

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