An integrating factor of the differential equation $\left(1-x^2\right) \frac{d y}{d x}+x y=\frac{x^4}{\left(1+x^5\right)}\left(\sqrt{1-x^2}\right)^3$ is

Let $y=y(x)$ be the solution of the differential equation $(1-x^2) dy = [xy + (x^3+2) \sqrt{3(1-x^2)}] dx$ for $-1 < x < 1$,with $y(0)=0$. If $y(1/2) = m/n$,where $m$ and $n$ are coprime numbers,then $m+n$ is equal to . . . . . . . . . .

The solution of the differential equation $x\frac{dy}{dx} + y = x^2 + 3x + 2$ is

If the differential equation $\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$ holds for all $x$ with initial conditions $f(0) = 1$ and $f'(0) = 2$, then which of the following is true?

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

Let $f(x)$ be differentiable on the interval $(0, \infty)$ such that $f(1)=1$,and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then $f(x)$ is