Express $\frac{dt}{dx} = \frac{t}{x + t e^{-2x/t}}$ in the form of $\frac{dx}{dt} = \phi\left(\frac{x}{t}\right)$.

The solution of $x \frac{dy}{dx} = y + x e^{y/x}$ with $y(1) = 0$ is

The slope of the tangent at $(x, y)$ to the curve passing through $(2, 1)$ is $\frac{x^2+y^2}{2xy}$. Find the equation of the curve.

$A$ bag contains $3$ red,$6$ white,and $7$ blue balls. Two balls are drawn one after another. What is the probability that the first ball is white and the second ball is blue,if the first ball drawn is not replaced in the bag?

Let $f(x) = \sqrt{\lim_{r \rightarrow x} \left\{ \frac{2r^2 \left[(f(r))^2 - f(x)f(r)\right]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}} \right\}}$ be differentiable in $(-\infty, 0) \cup (0, \infty)$ and $f(1) = 1$. Then the value of $ea$,such that $f(a) = 0$,is equal to:

If the gradient of the tangent at any point $(x, y)$ of a curve which passes through the point $\left( 1, \frac{\pi}{4} \right)$ is $\left\{ \frac{y}{x} - \sin^2\left( \frac{y}{x} \right) \right\}$,then the equation of the curve is: