The solution of the differential equation $(2x - 3y + 5)dx + (9y - 6x - 7)dy = 0$ is

The general solution of the differential equation $\frac{dy}{dx} = \cot x \cdot \cot y$ is

The equation of the curve passing through the point $(1, 1)$ such that the slope of the tangent at any point $(x, y)$ is equal to the product of its coordinates is

Let $y = y(x)$ be the solution curve of the differential equation $(1 + \sin x) \frac{dy}{dx} + (y + 1) \cos x = 0$ with the condition $y(0) = 0$. If the curve $y = y(x)$ passes through the point $(\alpha, -\frac{1}{2})$,then a value of $\alpha$ is:

The solution of the equation $\sin^{-1} \left( \frac{dy}{dx} \right) = x + y$ is

The particular solution of the differential equation $e^{\frac{dy}{dx}} = (x+1)$ with the condition $y(0) = 3$ is