The solution of the differential equation $(\sin x + \cos x)dy + (\cos x - \sin x)dx = 0$ is

The general solution of the differential equation $y y^{\prime} = x \left[ \frac{y^2}{x^2} + \frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi^{\prime}\left(\frac{y^2}{x^2}\right)} \right]$,where $\phi$ is an arbitrary function,is

If the curve $y = y(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \log_e x) dx$, $x > 0$, then $y(e)$ is equal to:

The equation of the curve passing through the point $(1,0)$ and whose slope is $\frac{y - 1}{x^2 + x}$ is

Find a particular solution satisfying the given condition:
$(x^{3}+x^{2}+x+1) \frac{dy}{dx} = 2x^{2}+x; y=1$ when $x=0$

Find the equation of the curve passing through the point $(1, 1)$ whose differential equation is $x dy = (2x^2 + 1) dx$ where $x \neq 0$.