If $\frac{d^2y}{dx^2} + \sin x = 0$,then the solution of the differential equation is ...... .

Suppose $f(x) = e^{ax} + e^{bx},$ where $a \neq b,$ and that $f''(x) - 2f'(x) - 15f(x) = 0$ for all $x.$ Then the product $ab$ is equal to

Let $\frac{dy}{dx} = \frac{ax - by + a}{bx + cy + a}$,where $a, b, c$ are constants,represent a circle passing through the point $(2, 5)$. Then the shortest distance of the point $(11, 6)$ from this circle is

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 $1$.

$A$ solution curve of the differential equation $(x^2+xy+4x+2y+4) \frac{dy}{dx}-y^2=0, x>0$,passes through the point $(1,3)$. Then the solution curve

The solution of $y{e^{ - x/y}}dx - (x{e^{ - x/y}} + {y^3})dy = 0$ is