The solution of the differential equation $\cos x \cos y \frac{dy}{dx} = - \sin x \sin y$ is

Given that the slope of the tangent to a curve $y=y(x)$ at any point $(x, y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$,then its equation is

If $y = y(x)$ is the solution of the differential equation $2x^{2} \frac{dy}{dx} - 2xy + 3y^{2} = 0$ such that $y(e) = \frac{e}{3}$,then $y(1)$ is equal to

The solution of the equation $\frac{dy}{dx} + \sqrt{\frac{1 - y^2}{1 - x^2}} = 0$ is

Show that the general solution of the differential equation $\frac{dy}{dx} + \frac{y^{2}+y+1}{x^{2}+x+1} = 0$ is given by $(x+y+1) = A(1-x-y-2xy)$,where $A$ is a parameter.

The solution of the differential equation $(x + y - 1)dx + (2x + 2y - 3)dy = 0$ is