The Integrating Factor of the differential equation $x \frac{dy}{dx} + 2y = x^2$ $(x \neq 0)$ is . . . . . . .

If the integrating factor of $x(1 - x^2)dy + (2x^2y - y - ax^3)dx = 0$ is $e^{\int Pdx}$,then $P$ is equal to

The particular solution of $e^{\frac{y}{x}} = x, y(1) = 3, x > 0$ is . . . . . . .

For the differential equation $y^2 dx + \left( x - \frac{1}{y} \right) dy = 0$ with the initial condition $y(1) = 1$,find $x$.

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:

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