Let $y=y(x)$ be the solution of the differential equation $(x^2+1) y^{\prime}-2 x y=(x^4+2 x^2+1) \cos x$,with $y(0)=1$. Then $\int_{-3}^3 y(x) d x$ is :

The solution of $r dx + (x - r^2) dr = 0$ is

By multiplying with $e^{\int P dx}$ on both sides of the equation $\frac{dy}{dx} + P(x)y = Q(x)$,the left side of the equation takes the form $\frac{d}{dx}(y f(x))$,then $f(x) =$

The solution of the differential equation $y' = y \tan x - 2 \sin x$ is

Let $f :(0, \infty) \rightarrow R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f^{\prime}(x)=2 x f(x)+3$,with $f(1)=4$. Then $2 f(2)$ is equal to:

If $x = x(y)$ is the solution of the differential equation $y \frac{dx}{dy} = 2x + y^{3}(y+1)e^{y}$ with the initial condition $x(1) = 0$,then $x(e)$ is equal to: