The differential equation of $y = A{e^{2x}} + B{e^{ - 2x}}$ is (where $A$ and $B$ are arbitrary constants):

Verify that the given function $x+y=\tan ^{-1} y$ is a solution of the differential equation $y^{2} y^{\prime}+y^{2}+1=0$.

The differential equation representing the family of ellipses having foci either on the $x$-axis or on the $y$-axis,centered at the origin,and passing through the point $(0,3)$ is:

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

The differential equation of the family of circles whose center lies on the $x$-axis is: