If $X = x + h, Y = y + k$ transforms $\frac{dy}{dx} = \frac{2x + 3y - 7}{3x + 2y - 8}$ to a homogeneous differential equation,then $(h, k) =$

The solution of the differential equation $3 x y' - 3 y + (x^2 - y^2)^{1/2} = 0$, satisfying the condition $y(1) = 1$ is

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

The solution of the differential equation $(x^2 + y^2)dx = 2xydy$ is

The solution of $x \frac{d y}{d x} = y(\log y - \log x + 1)$ is

The equation of a curve passing through $(1, 0)$ for which the product of the abscissa of a point $P$ and the intercept made by a normal at $P$ on the $x$-axis equals twice the square of the radius vector of the point $P$ is