Find a particular solution satisfying the given condition: $\cos \left(\frac{dy}{dx}\right) = a$ $(a \in R)$; $y = 1$ when $x = 0$.

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

If the solution of $\frac{dy}{dx} = (3x + y + 4)^2$ is $\frac{1}{\sqrt{3}} \tan^{-1}(f(x, y)) - x = k$, then $f(1, 2) = $

The solution of the differential equation $\frac{dy}{dx} = 1 + x + y + xy$ is

The equation of the curve passing through the point $(0, -2)$ given that at any point $(x, y)$ on the curve,the product of the slope of its tangent and the $y$-coordinate of the point is equal to the $x$-coordinate of the point,is

General solution of the differential equation $(y^3+y)(x^2+1) dy = (xy^4+2y^2x) dx$ is (where $C$ is a constant of integration.)