Let $f : R \rightarrow R$ be a differentiable function with $f(0)=1$ and satisfying the equation $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in R$. Then,the value of $\log _e(f(4))$ is:

Find the general solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ where $x \neq 0$.

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2 \sin x-5)x-2 \cos x$ such that $y(0)=7$. Then $y(\pi)$ is equal to :

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

The general solution of the differential equation $\frac{dy}{dx} = y \tan x - y^2 \sec x$ is

Let $y$ be the solution of the differential equation $(1-x^{2}) dy = (xy + (x^{3}+2) \sqrt{1-x^{2}}) dx$ for $-1 < x < 1$ with $y(0)=0$. If $\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) dx = k$,then $k^{-1}$ is equal to: