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

To reduce the differential equation $\frac{dy}{dx} + P(x)y = Q(x)y^n$ to the linear form,the substitution is

Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10 \int_0^x (t+2) f(t) dt$ for $x \geq 0$. Then $f(2)$ is equal to . . . . . .

Let $f$ be a differentiable function $f : R \rightarrow R$ satisfying the equation $f(x) = (1+x^2) \left[ 1 + \int_{0}^{x} \frac{f(t)}{1+t^2} dt \right]$ for all $x \in R$. Then $f(1)$ is:

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 $S$?

General solution of the differential equation $\frac{dy}{dx} + y \tan x = \sec x$ is