If $y = (A + Bx) e^{mx} + (m - 1)^{-2} e^x$,then $\frac{d^2y}{dx^2} - 2m \frac{dy}{dx} + m^2y$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\sec x \, dy + \{2(1-x) \tan x + x(2-x)\} \, dx = 0$ such that $y(0)=2$. Then $y(2)$ is equal to :

If $y'' - 3y' + 2y = 0$ where $y(0) = 1$ and $y'(0) = 0$, then the value of $y$ at $x = \log_{e} 2$ is

The solution of $\frac{dy}{dx} + \frac{1}{3}y = 1$ is

Let $y^{\prime}(x) + y(x) g^{\prime}(x) = g(x) g^{\prime}(x)$,$y(0) = 0$,$x \in \mathbb{R}$,where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb{R}$ with $g(0) = g(2) = 0$. Then the value of $y(2)$ is

The integrating factor of $x \frac{dy}{dx} - 2y = x^2 + \sin \left( \frac{1}{x^2} \right)$ is