$A$ function $y = f(x)$ satisfies the differential equation $f(x) \sin 2x - \cos x + (1 + \sin^2 x) f'(x) = 0$ where $f(0) = 0$. Then the value of $f(\frac{\pi}{6})$ is equal to

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 $f : (0, \infty) \to (2, 20)$ be a twice differentiable function such that $\lim_{x \to \infty} (f(x) + f'(x) + f''(x)) = \lim_{x \to \infty} g(x)$,where $\lim_{x \to \infty} g(x)$ exists and is equal to $5$. Then $\lim_{x \to \infty} (f(x) - g(x))$ is equal to:

Find the general solution of the differential equation: $\frac{dy}{dx} + 3y = e^{-2x}$

An integrating factor of the differential equation $x \frac{dy}{dx} + y \log x = x e^x x^{-\frac{1}{2} \log x}$,$(x > 0)$ is

Let $y=y(x)$ be the solution of the differential equation $(2x \ln x) \frac{dy}{dx} + 2y = \frac{3}{x} \ln x$,$x > 0$ and $y(e^{-1}) = 0$. Then,$y(e)$ is equal to