The differential equation for which $\sqrt{1+y^2}=C x e^{\tan ^{-1} x}$ is the general solution, is

Let $c_1, c_2, c_3, c_4$ be arbitrary constants. The order of the differential equation,corresponding to $y=c_1 e^x+c_2 e^{\log _{e} x}+c_3 \sin ^2 x-c_4\left(\cos ^2 x-1\right)$ is

Verify that the given function $y = \cos x + C$ is a solution of the differential equation $y^{\prime} + \sin x = 0$.

The differential equation whose solution is $(x - h)^2 + (y - k)^2 = a^2$ is ($a$ is a constant).

The differential equation representing the family of circles having their centres on the $Y$-axis is (where $y_1 = \frac{dy}{dx}$ and $y_2 = \frac{d^2y}{dx^2}$):

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