The integrating factor of the differential equation $(\tan ^{-1} y-x) dy = (1+y^2) dx$ is . . . . . . .

The equation of the curve passing through the point $(0, \pi)$ and satisfying the differential equation $y dx = (x + y^3 \cos y) dy$ is

Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$,where $\alpha > 0, \beta > 0$ and $\gamma > 0$. Then $\lim_{t \rightarrow \infty} y(t)$ is:

Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10 \int_0^x (t+2) f(t) dt$ for $x \geq 0$. Then $f(2)$ is equal to . . . . . .

The general solution of a differential equation of the type $\frac{dx}{dy} + P_{1}x = Q_{1}$ is

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+2y=\sin(2x)$ with $y(0)=\frac{3}{4}$,then $y\left(\frac{\pi}{8}\right)$ is equal to: