Let $y=y(x)$ be the solution of the differential equation $(x^2+4)^2 dy + (2x^3y+8xy-2) dx = 0$. If $y(0)=0$,then $y(2)$ is equal to

If a curve passes through the point $(1, -2)$ and has the slope of the tangent at any point $(x, y)$ on it as $\frac{x^2 - 2y}{x}$,then the curve also passes through the point:

For any real numbers $\alpha$ and $\beta$,let $y_{\alpha, \beta}(x), x \in R$,be the solution of the differential equation $\frac{dy}{dx}+\alpha y=x e^{\beta x}, y(1)=1$. Let $S=\{y_{\alpha, \beta}(x): \alpha, \beta \in R\}$. Then which of the following functions belong$(s)$ to the set $S$?

Let $f$ be a differentiable function with $\lim _{x \rightarrow \infty} f(x)=0$. If $y^{\prime}+y f^{\prime}(x)-f(x) f^{\prime}(x)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$, then:

The solution of the differential equation $2x \frac{dy}{dx} - y = 3$ represents a family of

Let $y=f(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^6+4x}{\sqrt{1-x^2}}$ for $-1 < x < 1$ such that $f(0)=0$. If $6 \int_{-1/2}^{1/2} f(x) dx = 2\pi - \alpha$,then $\alpha^2$ is equal to . . . . . .