The integrating factor of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is:

The integrating factor of the linear differential equation $\frac{dy}{dx} = \frac{1}{4x + 3y}$ 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 . . . . . . . . . .

An integrating factor of the equation $(1+y+x^2 y) dx+(x+x^3) dy=0$ is

Let $f: [1, \infty) \rightarrow R$ be a differentiable function such that $f(1) = \frac{1}{3}$ and $3 \int_1^x f(t) dt = x f(x) - \frac{x^3}{3}$ for $x \in [1, \infty)$. Then the value of $f(e)$ is:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)((y+1)e^{x^{2}/2}-x)$,with $y(2)=0$. Then $y'(1)$ is equal to . . . .