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

The general solution of $\frac{dy}{dx} = x + \sin x \cos y + x \cos y + \sin x$ is

Find the particular solution of the differential equation $\log \left(\frac{d y}{d x}\right)=3 x+4 y$ given that $y=0$ when $x=0$.

Let a curve $y=f(x)$ pass through the point $(2, (\ln 2)^2)$ and have slope $\frac{2y}{x \ln x}$ for all positive real values of $x$. Then the value of $f(e)$ is equal to:

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$. If the curve passes through the points $(1, 2)$ and $(8, 1)$,then $\left| y \left(\frac{1}{8}\right) \right|$ is equal to

Show that the general solution of the differential equation $\frac{dy}{dx} + \frac{y^{2}+y+1}{x^{2}+x+1} = 0$ is given by $(x+y+1) = A(1-x-y-2xy)$,where $A$ is a parameter.