If $\frac{d^2y}{dx^2} = 0$,then:

The differential equation whose solution is $y = c_{1} \cos(ax) + c_{2} \sin(ax)$ (where $c_{1}$ and $c_{2}$ are arbitrary constants) is

If $m$ and $n$ are the order and degree of the differential equation of the family of parabolas with focus at the origin and $X$-axis as its axis, then $m n-m+n=$

Verify that the given function $y=ae^{x}+be^{-x}+x^{2}$ is a solution of the differential equation $x \frac{d^{2} y}{dx^{2}}+2 \frac{dy}{dx}-xy+x^{2}-2=0$.

$y = ax + b$ is

The differential equation of the family of curves $r^2 = a^2 \cos 2\theta$,where '$a$' is an arbitrary constant,is: