The general solution of the differential equation $\frac{dy}{dx} = e^{x-y}$ is . . . . . .

The equation of a curve passing through the point $(0,1)$,given that the slope of the tangent to the curve at any point $(x, y)$ is equal to the sum of the $x$-coordinate and the product of $x$ and $y$ coordinates at that point,is

General solution of $(x+y)^{2} \frac{d y}{d x}=a^{2}, a \neq 0$ is ($C$ is an arbitrary constant)

If $\frac{a}{a_1}=\frac{b}{b_1}$,then the substitution to be used to solve the differential equation $\frac{dy}{dx}=\frac{ax+by+c}{a_1x+b_1y+c_1}$ by using separation of variables is

Find the equation of the curve passing through the point $\left(0, \frac{\pi}{4}\right)$ whose differential equation is $\sin x \cos y \, dx + \cos x \sin y \, dy = 0$.

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