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:

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 $\frac{dy}{dx} + y = 1$ is

The solution of the differential equation $(1 + y^2) + (x - e^{\tan^{-1}y}) \frac{dy}{dx} = 0$ is

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and $\lim _{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1$ for each $x>0$. Then,for all $x>0, f(x)$ is equal to

Let a curve $y = y(x)$ pass through the point $(3,3)$ and the area of the region under this curve,above the $x$-axis and between the abscissae $3$ and $x (>3)$ be $\left(\frac{y}{x}\right)^{3}$. If this curve also passes through the point $(\alpha, 6\sqrt{10})$ in the first quadrant,then $\alpha$ is equal to $........$