The differential equation of the family of ellipses $\frac{x^2}{a^2} + \frac{y^2}{b^2} = c$ is given by $\left( y' = \frac{dy}{dx}, y'' = \frac{d^2y}{dx^2} \right)$.

Which of the following differential equations has $y=x$ as one of its particular solutions?

If the order of a differential equation $\frac{d^2 y}{d x^2}-2\left(\frac{d y}{d x}\right)^3+\sin \left(\frac{d y}{d x}\right)+y=0$ is $l$ and the degree of the differential equation $\left(1+\frac{d^2 y}{d x^2}\right)^{\frac{2}{3}}=\left[2-\left(\frac{d y}{d x}\right)^3\right]^{\frac{3}{2}}$ is $m$, then the differential equation corresponding to the family of curves $y=A x^l+B e^{m x}$, where $A$ and $B$ are arbitrary constants, is

The normal form of the equation of a line is $x \cos \alpha + y \sin \alpha = p$. Find the differential equation of the family of all such lines, where $p$ and $\alpha$ are arbitrary constants.

The differential equation obtained by eliminating the arbitrary constants from the equation $y^{2}=(2 x+c)^{5}$ is

The differential equation of the family of curves $y = a e^x + b x e^x + c x^2 e^x$,where $a, b, c$ are arbitrary constants,is