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

The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx} + \frac{x+a}{y-2} = 0, y(1) = 0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect the $x$-axis at points $R$ and $S$ respectively,then the length of the line segment $RS$ is

Number of straight lines which satisfy the differential equation $\frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0$ is:

The equation of a curve passing through $\left( 2, \frac{7}{2} \right)$ and having gradient $1 - \frac{1}{x^2}$ at $(x, y)$ is:

The population of towns $A$ and $B$ increases at a rate proportional to their population present at that time. At the end of the year $1984$,the population of both the towns was $20,000$. At the end of the year $1989$,the population of town $A$ was $25,000$ and that of town $B$ was $28,000$. The difference of populations of towns $A$ and $B$ at the end of $1994$ was

If the solution of the differential equation $\frac{dy}{dx} = \frac{1+x}{2y}$ is a conic passing through the point $(1, 1)$,then its eccentricity is: