The solution of the differential equation $e^x y dx + e^x dy + x dx = 0$ is

$A$ curve passing through $(2, 3)$ and satisfying the differential equation $\int\limits_0^x {t\,y(t)\,dt} = x^2y(x)$ for $x > 0$ is

Let $f$ be a non-negative function defined on the interval $[0,1]$. If $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ for $0 \leq x \leq 1$ and $f(0)=0$,then:

The solution of $\log \left( \frac{dy}{dx} \right) = ax + by$ is

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

$A$ particle starts at the origin and moves along the $x$-axis in such a way that its velocity at the point $(x, 0)$ is given by the formula $\frac{dx}{dt} = \cos^2(\pi x)$. Then the particle never reaches the point on: