Let $x=x(y)$ be the solution of the differential equation $2 y e^{x / y^{2}} d x+\left(y^{2}-4 x e^{x / y^{2}}\right) d y=0$ such that $x(1)=0$. Then,$x(e)$ is equal to

Let a differentiable function $f$ satisfy $f(x) + \int_{3}^{x} \frac{f(t)}{t} dt = \sqrt{x+1}$ for $x \geq 3$. Then $12f(8)$ is equal to:

Let $\alpha x = \exp(x^\beta y^\gamma)$ be the solution of the differential equation $2x^2 y \frac{dy}{dx} - (1 - xy^2) = 0$,for $x > 0$ and $y(2) = \sqrt{\log_e 2}$. Then $\alpha + \beta - \gamma$ equals:

The integrating factor of the differential equation $x \frac{dy}{dx} + 2y = x^2 \log x$ is

Find the particular solution of the following differential equation,given that $y=1$ when $x=0$: $(1+x^2) \frac{dy}{dx} = e^{\tan^{-1} x} - y$.

Let $y = y(x)$ be the solution of the differential equation: $\frac{dy}{dx} + \left( \frac{6x^2 + (3x^2 + 2x^3 + 4)e^{-2x}}{(x^3 + 2)(2 + e^{-2x})} \right) y = 2 + e^{-2x}, x \in (-1, 2)$, satisfying $y(0) = \frac{3}{2}$. If $y(1) = \alpha(2 + e^{-2})$, then $\alpha$ is equal to: