Let $y=y_{1}(x)$ and $y=y_{2}(x)$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$,with $y_{1}(0)=0$ and $y_{2}(0)=1$ respectively. Then,the number of points of intersection of $y=y_{1}(x)$ and $y=y_{2}(x)$ is.

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the tangent line $Y-y=Y^{\prime}(x)(X-x)$ and the coordinate axes,where $(x, y)$ is any point on the curve,is always $\frac{-y^2}{2 Y^{\prime}(x)}+1$,where $Y^{\prime}(x) \neq 0$. If $Y(1)=1$,then $12 Y(2)$ equals

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

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

Find the general solution of the differential equation: $x \frac{dy}{dx} + 2y = x^2 \log x$.

Find a particular solution satisfying the given condition: $\frac{dy}{dx} - 3y \cot x = \sin 2x$; $y = 2$ when $x = \frac{\pi}{2}$.