The solution for the differential equation $\frac{dy}{y} + \frac{dx}{x} = 0$ is:

The solution of $\tan y \frac{dy}{dx} = \sin(x+y) + \sin(x-y)$ is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function which satisfies $f(x) = \int_0^x f(t) \, dt$. Then the value of $f(\ln 5)$ is

The solution of the differential equation $x \frac{dy}{dx} - y = 0$ is . . . . . . .

If the solution $y(x)$ of the given differential equation $(e^y+1) \cos x \, dx + e^y \sin x \, dy = 0$ passes through the point $(\frac{\pi}{2}, 0)$,then the value of $e^{y(\frac{\pi}{6})}$ is equal to ...........

For the differential equation $x y \frac{dy}{dx} = (x+2)(y+2)$,find the solution curve passing through the point $(1, -1)$.