If $\frac{dy}{dx} + y \tan x = \sin 2x$ and $y(0) = 1$,then $y(\pi)$ is equal to

If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $\left(7 x^4 \cot y-e^x \operatorname{cosec} y\right) \frac{d x}{d y}=x^5, x \geq 1$,then at $x=2$,the value of $\cos y$ is:

If the solution of the differential equation $x \frac{dy}{dx} + y = x e^x$ is $xy = e^x \phi(x) + C$,then $\phi(x)$ 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

The general solution of the differential equation $\frac{dy}{dx} + (\sec x \operatorname{cosec} x) y = \cos^2 x$ is

Let $y=y(x)$ be the solution of the differential equation $x \log _e x \frac{d y}{d x}+y=x^2 \log _e x, (x > 1)$. If $y(2)=2$,then $y(e)$ is equal to