$A$ function $y = f(x)$ satisfies the condition $f'(x) \sin x + f(x) \cos x = 1$,where $f(x)$ is bounded as $x \rightarrow 0$. If $I = \int_{0}^{\frac{\pi}{2}} f(x) \, dx$,then:

Statement $-1$: The slope of the tangent at any point $P$ on a parabola,whose axis is the $x$-axis and vertex is at the origin,is inversely proportional to the ordinate of the point $P$.
Statement $-2$: The system of parabolas $y^2 = 4ax$ satisfies a differential equation of degree $1$ and order $1$.

If $\phi(x) = \frac{1}{\sqrt{x}} \int \limits_0^x (4 \sqrt{2} \sin t - 3 \phi^{\prime}(t)) dt, \quad x > 0$,then $\phi^{\prime}\left(\frac{\pi}{4}\right)$ is equal to:

The solution of the differential equation $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ is

$A$ continuously differentiable function $\phi (x)$ in $(0, \pi)$ satisfying $y' = 1 + y^2$ and $y(0) = 0 = y(\pi)$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{\sin y + e^x}{\ln y - x \cos y}$ is: