Let $A$ and $B$ be two independent events. If $P(A) = \frac{1}{5}$ and $P(A \cup B) = \frac{7}{10}$,then what is the value of $P(\overline{B})$?

The slope of the tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2+y^2}{2xy}$,where $x > 0$. If $y(2) = 0$,then a value of $y(8)$ is

If $\frac{dy}{dx} = \frac{xy}{x^2 + y^2}$ and $y(1) = 1$,then find the value of $x$ that satisfies $y(x) = e$.

For the differential equation,the general solution for $x \cos \left( \frac{y}{x} \right) (y dx + x dy) = y \sin \left( \frac{y}{x} \right) (x dy - y dx)$,(where $c$ is the constant of integration) is

The general solution of $y^2 dx + (x^2 - xy + y^2) dy = 0$ is:

The slope of the tangent at $(x, y)$ to a curve passing through $\left( 1, \frac{\pi}{4} \right)$ is given by $\frac{y}{x} - \cos^2\left( \frac{y}{x} \right)$. Then the equation of the curve is: