The integrating factor of the differential equation $(1 + t^2) + (x - e^{\tan^{-1} t}) \frac{dt}{dx} = 0$ is

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:

The solution of the differential equation $\frac{dx}{dy} = \frac{x}{1 + x e^y \cos(x^2)}$ is (where $c$ is the constant of integration):

The particular solution of the differential equation $\sin^{2} y \frac{dx}{dy} + x = \cot y$ when $x = 0$ and $y = \frac{3\pi}{4}$ is

$A$ function $f(x)$ satisfies the condition $f(x) = f'(x) + f''(x) + f'''(x) + \dots \infty$,where $f(x)$ is an indefinitely differentiable function and the dash denotes the order of the derivative. If $f(0) = 1$,then $f(x)$ is:

Let $y=y(x)$ be the solution of the differential equation $x^{4}dy + (4x^{3}y + 2\sin x)dx = 0$, $x>0$, $y(\frac{\pi}{2})=0$. Then $\pi^{4}y(\frac{\pi}{3})$ is equal to: