The solution of the differential equation $x \frac{dy}{dx} + y = x^3 y^6$ is:

The equation of the curve passing through the origin and satisfying the equation $(1 + {x^2})\frac{{dy}}{{dx}} + 2xy = 4{x^2}$ is

Let $\alpha x = \exp(x^\beta y^\gamma)$ be the solution of the differential equation $2x^2 y \frac{dy}{dx} - (1 - xy^2) = 0$,for $x > 0$ and $y(2) = \sqrt{\log_e 2}$. Then $\alpha + \beta - \gamma$ equals:

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

If $x^2 y - x^3 \frac{dy}{dx} = y^4 \cos x$,then $x^3 y^{-3}$ is equal to

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.